Graph Theory (draft)

arXiv:1102.1087v1 [cs.DM] 5 Feb 2011

GRAPH THEORY JESSE D. GILBERT 4 Feb 2011

Chapter 1

An abbreviated higher graph theory 1.1

Graphic sequences

A degree sequence is a list of integers with multiplicity. If a degree sequence π is realizable, there is a graph G whose list of vertex degrees compose the degree sequence π. Degree sequences are not always realizable, particularly if the degree sequence has odd sum. The existence of graphic sequences possessing combinations of properties has been studied and we hope to apply the results of those studies in some way. It is worthwhile then to examine the nature of graphic sequences and the machinery we use to explore and manipulate graphic sequences. Realization of Sequences. Different types of graphs have different rules for the feasibility of realization. Any balanced sequence has a realization as a multi-digraph. Because loops are allowed, as are 2-cycles, this is not a difficult result to prove. There has been some attempt to enumerate the number of realizations of a multi-digraph up to degree sequence or number of arcs using induction. Directed graphs with the same degree sequence can be obtained from one another by a series of swaps as well. The smallest set of swaps that suffice to transform one realization to another realization has not yet been characterized given the restriction that no intermediary graph in the sequence can have loops or multiple arcs. A triangle swap on a digraph is a swap that preserves digraphic degree sequences and exchanges a directed P3 for a loop and a 2-loop. The ~1 ∪ K ~2 . triangle swap is denoted as P~3 → K If intermediary digraphs with loops and multiple arcs allowed, the triangle swap is sufficient alone to transform a digraph D into any other digraph D′ whenever π(D) = π(D′ ). Undirected graphs with the same degree sequence can be obtained from each other by a series of C4 -swaps or C2k -swaps. There are even more exotic swaps which act on (edge, non-edge)-alternating eulerian cycles. It is never the case

1

that a graph with degree sequence π and a graph with a degree sequence π ′ can be obtained from one another by C4 -swaps unless π = π ′ . The value σ(G, n) is the smallest value such that every degree sequence π with degree sum σ(G, n) potentially contains a copy of G. Theorem 1.1.1. [Erdos-Gallai] A sequence π: d1 ≥ d2 ≥ ... ≥ dp of nonnegative integers whose sum (say s) is even, is graphic if and only if i=k X i=1

di ≤ k(k − 1) +

j=p X

j=k+1

{dj , k}

for every k such that 1 ≤ k ≤ p. Theorem 1.1.2. [Havel-Hakimi] For n > 1 a non-increasing integer list π = d1 , d2 ...dp with p terms is graphic if and only if π ′ is graphic where π ′ is obtained from π by deleting its largest element d1 and subtracting 1 from its d1 next largest terms. The only 1-term graphic sequence is d1 = 0. There have been some recent results about graphic sequences [34]. Pi=n Corollary 1.1.3. If π = (d1 , d2 , ..., dn ) is non-increasing, m = d1 and i=1 di 2

is even, and if there exists n1 ≤ n such that dn1 ≥ h ≥ 1 and n1 ≥ h1 [ (m+h+1) ], 4 then π is graphic. We use Erdos-Gallai to prove π is graphic. Since n1 ≥

1 1 1 (m + h + 1)2 [ ] ≥ (mh + 1) = m + , h 4 h h

we have n1 ≥ m + 1. By Erdos-Gallai, we can verify π is graphic in 3 steps: Suppose 1 ≤ t ≤ h. Suppose h < t ≤ m. Suppose m < t ≤ n − 1. Problems. 1. Prove that if S(k,m) is the set of (k, m)-graphs up to isomorphism, the total number of m-edge graphs up to isomorphism is counted by 

2m 2

m




=

n=2m X n=2

(n2 )
m
n |S(k,m) |. k

2. Find a 2r-regular graph with no (2r − 1)-factor for each choice of r. 3. Classify the degree sequences of the trees.

1.2

Connectivity

One of the most important theorems in the study of connectivity is Menger’s Theorem. This theorem’s statement relies on the definition of the parameter kconnectivity. In graph theory and digraph theory, the most sophisticated notions 2

of strongly connected, hamiltonian-connected, and traceable for digraphs in particular have applications. A graph G is k-connected if G − S is connected for all S ⊂ V (G) where |S| ≤ k − 1. A graph G is k-edge-connected if G − S is connected for all S ⊂ E(G) where |S| ≤ k − 1. Theorem 1.2.1. [Menger] If x, y are vertices of a graph G and xy 6∈ E(G), then the minimum size of an (x, y)-cut equals the maximum number of pairwise internally disjoint (x, y)-paths. Theorem 1.2.2. If G is k-connected, then for all S ⊂ V (G) with |S| ≤ k, there is a cycle Ct ⊂ G such that S ⊂ V (Ct ). The edge-connectivity of a graph G, κ1 (G), is the minimum number of edges whose removal disconnects the graph G. For all graphs we, have κ(G) ≤ κ1 (G) ≤ δ(G). Theorem 1.2.3. If δ(G) ≥ ⌊ n2 ⌋, then the edge-connectivity of G is exactly δ(G). The term k-linked into used to describe the condition that a k-linked graph G has k vertex-disjoint (u, v)-paths between any two vertices u, v ∈ V (G). Theorem 1.2.4. If G is 2k-connected and has girth 11, then G is k-linked for k > 5. If k = 4, 5, then if G is 2k-connected and has girth 19, then G is k-linked. Problems. 1. Prove Menger’s theorem. 2. Find a sufficient condition for a magnanimous graph to be hamiltonian. 3. Suppose the girth of G is k. Find r that fixes G hamiltonian if G is rregular. 4. Prove Grinberg’s theorem. See [49].

1.3

Adjacency matrices

The central definitions of graph theory include adjacency, order, and size, all of which can be detected by the matrix approach to graphs. Walks. Eulerian circuits EG are E(G)-spanning circuits of graphs G. The subject of eulerian graphs is almost synonymous with the inception of graph theory. A walk is a list v1 , e1 , v2 , ..., ek−1 , vk of vertices and edges such that for all 1 ≤ i ≤ k, the edge ei is the unordered pair vi vi+1 . The length of a walk is the number of edges in the walk. Walks are often abbreviated by just listing the order of vertices. A trail is a walk with no repeated edge. A circuit is a closed trail; that is a circuit is a trail whose initial and terminal vertices are identical. An eulerian circuit C is a circuit in G such that E(G) ⊂ E(C). A graph G is eulerian if and only if the degree of every vertex in G is even and the graph is connected. It is not always trivial 3

to build an eulerian circuit in a graph, even once it is obvious the graph is eulerian. However, there is a simple algorithm that achieves the construction in every case. A graph that is eulerian has a decomposition into cycles. Definition 1 The adjacency matrix of G, M (G) is an n × n square matrix which has an entry at the uv and vu coordinate if and only if u ∼ v in G, that is, if uv ∈ E(G). The powers of M (G), M (G), M 2 (G), M 3 (G), ... are just the powers of M (G) under standard matrix multiplication. Theorem 1.3.1. The number of uv-walks of length j in G is given by the uv-coordinate of M j (G). Hamiltonicity. A uv-path is a walk from u to v which is a subgraph of G such that no vertex or subsequence of two vertices repeats. A path of order n, Pn , is a tree with degree sequence (1)2 , (2)n−2 . A cycle of order n, Cn , is the unique connected simple graph with degree sequence (2)n . A tree Tn is a simple graph which has no cycles as subgraphs, but which has n(G) − 1 = m(G). We say a subgraph H of G spans G if V (G) ⊂ V (H). A graph is connected if there is a path from u to v for all (u, v) pairs in the vertex set, and consequently, a graph is connected if and only if it has a spanning tree. A component of G is a vertexmaximal connected subgraph of G. A hamiltonian cycle is a spanning cycle of G. A hamiltonian path is a spanning path of G. If a graph G is traceable, it has a spanning path. A pancyclic graph has a cycle of every order and a panarboreal graph of order n contains every tree of of order n. Edge-pancyclic and vertex-pancyclic graphs are graphs that respectively contain every edge or vertex on a cycle of every length. To some extent, the hamiltonian cycle problem is related to the cyclesubgraph problem. If G is vertex-pancyclic, for instance, every vertex is on an induced hamiltonian subgraph of every order ≤ n(G). Hamiltonian-connected graphs have every two vertices as leaves on a spanning path. The terms cycle-extendible and cycle-reducible refer to the respective conditions that for any cycle in a graph there exists a vertex off the cycle that can be added to the vertex set of the cycle to form a cycle on this larger set of vertices, or, there exists a vertex on the cycle, that can be deleted from the vertex set such that another cycle on the reduced vertex set is in the edge set. It is important to observe that any uv-walk contains a uv-path. A graph G is k-connected if G − S is connected for all S ⊂ V (G) where |S| ≤ k − 1. A graph G is k-edge-connected if G − S is connected for all S ⊂ E(G) where |S| ≤ k − 1. Let κ(x, y) denote the minimum size of a set of vertices that interect every xy-path in G. Then let λ(x, y) be the maximum number of pairwise internally disjoint xy-paths in G. Theorem 1.3.2. [Menger] If x, y are vertices of a graph G and xy 6∈ E(G), then the minimum size of an x, y-cut equals the maximum number of pairwise internally disjoint x, y-paths. That is, λ(x, y) = κ(x, y). Theorem 1.3.3. If G is k-connected, then for all S ⊂ V (G) with |S| ≤ k, there is a cycle Ct ⊂ G such that S ⊂ V (Ct ). Toughness was introduced as an attempt to give a characterization of the hamiltonian graphs. The hamiltonian graphs are 1-tough. The toughness of a graph 4

t0 (G) = min|S|/c(G − S) where here c(G − S) is the number of components of G − S where S ⊂ V (G). Theorem 1.3.4. If δ(G) ≥ ⌊ n2 ⌋, then the edge-connectivity of G is exactly δ(G). Theorem 1.3.5. For all G with edge-connectivity κ1 (G), suppose that for all t such that 1 ≤ t ≤ min{n/2 − 1, δ} we have i=t X i=1

(di + dn−i ) ≥ tn − 1

where here π(G) = d1 , d2 , ..., dn . It follows that δ(G) = κ1 (G). There are strong analogues to these theorems for digraphs. Generally, the denser a graph is, the harder it is to disconnect a graph by removing vertices or edges. This has the interesting effect that the graph also becomes easier to dominate in extreme cases. That is, as a graph has more and more edges, we can generally pick a smaller and smaller subset of vertices that forms a subset of the vertex set such that V (G) − S ⊂ N (S). Domination. The parameter γ(G) is size of the smallest set of vertices S such that S ∪N (S) = V (G). The parameter γ(G) is known as the domination number of G and γi (G) is the independent domination number of G, which is defined identically, except that we require that the set S be an independent set. Let G be a graph of order n with δ = δ(G) ≥ 2. Then γ(G) ≤

n(1 + ln(δ + 1) . δ+1

and take a random set s ⊂ V (G) whoses vertices are chosen Set p = ln(δ+1) δ+1 independently with probability p. That is, consider the probability space whose elements are the 2n subsets of V (G). For a random set S, let S ′ be the vertices v 6∈ S such that if w ∈ N (v), w 6∈ S. Then S ′ ∪ S is a dominating set of G. The expected value E(|S| + |S ′ |) = np + ne−p(δ+1) =

n(1 + ln(δ + 1)) . δ+1

Problems. 1. Find the number of k-cycles in Km,n for all k. 2. Find the number of k-walks in Km,n for all k. 3. Prove that if G has 2k odd valence vertices, then G can be decomposed into cycles and k paths. 4. Prove K2n+1 has a hamiltonian cycle decomposition. 5

1.4

Distance

The distance dG (u, v) is the length of the shortest path between the vertices u and v in the graph G. The eccentricity of v is the value ecc(v) = maxu∈V (G) dG (u, v). The largest eccentricity of any vertex in G is the diameter of G, diam(G), while the smallest eccentricity of any vertex in G is the radius of G, rad(G). Theorem 1.4.1. For all graphs G, we have rad(G) ≤ diam(G) ≤ 2rad(G). The center of G, H = cen(G) is the subgraph in G induced by the set of vertices such that ecc(v) = rad(G). The periphery of G, H ′ = per(G) is the subgraph in G induced by the set of vertices such that ecc(v) = diam(G). Problems. 1. Find the set of graphs such that H = cen(G) for some G. Find the set of graphs such that H = per(G) for some G. 2. Find a graph of diameter n and radius n − ∆(G). 3. Consider the set of graphs such that if |{x : dG (v, x) = k}| = tx,k , then tx,k = tv,k for all v ∈ V (G). Call this set of graphs the magnanimous graphs. Show that for a magnanimous graph G, per(G) = cen(G). 4. Find two non-isomorphic magnanimous graphs of radius k for all k.

1.5

Matchings

Theorem 1.5.1. [Tutte’s] For all graphs G has a 1-factor if and only if S ⊂ V (G) has the number of odd components in [G − S] ≤ |S|. Theorem 1.5.2. [Konig-Egervary] The maximum cardinality of a matching in G is equal to the minimum cardinality of a vertex cover of its edges. Theorem 1.5.3. [Hall’s] The graph G contains a matching of A if and only if |N (S)| ≥ |S| for all S ⊂ A. The Hall theorem and the Konig-Egervary theorem can be used to prove each other as a corollary. Problems. 1. Prove Hall’s theorem and the Konig-Egervary theorem. 2. Show every 2r-regular graph has a 2-factorization. 3. Show that if G is 2-connected, then there exists t0 such that every t0 -tough graph has a 2-factor. See [49]. 6

1.6

Factors

Conjecture 1.6.1. [Akiyama] The linear arboricity of any r-regular graph is ⌈(r + 1)/2⌉. The conjecture holds for r = 2, 3, 4, 5, 6. Theorem 1.6.2. [Hajnal-Szemeredi] If G has order n and chromatic number χ, it is necessarily a subgraph of a complete near-balanced multiparite graph Tn,χ . Theorem 1.6.3. For every ǫ > 0 and for every positive integer h, there exists an n0 = n0 (ǫ, h) such that for every graph H with h vertices and for every n > n0 , any graph G with hn vertices and with minimum degree δ(G) ≥ (1 − 1/χ(H) + ǫ)hn has an H-factor. Let k0 (G) be the number of odd-order components of G. Theorem 1.6.4. [Tutte] For all finite graphs G, the graph G has a 1-factor if and only if for all S ⊂ V (G), k0 (G − S) ≤ |S|. Generally, it is difficult to show a graph has a 1-factor, even with Tutte’s Theorem as a lemma. Some weakenings of the condition include {P2 , P3 }-factors. The next four results and the following conjecture can all be found in [2]. Theorem 1.6.5. Every graph with ∆(G)/δ(G) ≤ 2 has a {P2 , P3 }-factor. Theorem 1.6.6. Every maximal planar graph has a {P2 , P3 }-factor. Theorem 1.6.7. The graph G has a 1-factor whenever 2|n(G), κ(G) = r and κ1 (G) = r − 1. Theorem 1.6.8. Let G be 3-edge connected, cubic graph of order 4p. Then for any two edges, e1 , e2 , there is a P4 -factor of G which contains both edges. Theorem 1.6.9. [Enomoto, Jackson, Katernis, Saito] Let k be a positive integer and G a graph. If G is k-tough, |V (G)| ≥ k + 1 and k|E(G)| is even, then G has a k-factor. For any positive ǫ, there is a (k − ǫ)-tough graph with no k-factor. Theorem 1.6.10. If G is [r, r +1]-regular and 1 ≤ k ≤ r, then G has a [k, k +1]factor. Problems. 1. Show there is a non-hamiltonian cubic planar graph. 2. For m ≥ 2, if m(G) ≥ m(n − 1)/2 has a cycle of length more than m. 4. Prove Dirac’s theorem. 5. Give an example of a graph that has full cycle spectrum except for an (n − 1)-cycle. 6. Show there is a 4-regular hamiltonian graph with no hamiltonian decomposition. 7. Find G such that GL = G or show no such G exists.

7

Chapter 2

Full introduction to graph theory The following is intended as a second introduction to graph theory.

2.1

Glossary of graphs

Graph theory has applications in areas where mathematical analysis intersects with the implementation of a model. The following definitions and remarks should provide enough information to make a good start on the problems that form the foundation of graph theory. Most of the information is limited to the classical results. The focus here is on general structural observations. A simple graph G is a set of two sets V and E where E is a set of unordered pairs of elements from the vertex set. The order n(G) of a graph is the cardinality of V . An isomorphism from a simple graph G to a simple graph H is a bijection f : V (G) −→ V (H) such that uv ∈ E(G) if and only if f (u)f (v) ∈ E(H). We say G is isomorphic to H if there exists such a function f . A simple graph H is a subgraph of G if there exists a subset of V (G) and a subset of E(G) when taken together as a graph G′ that is isomorphic to H. We write H ⊂ G if H is a subgraph of G. The neighborhood of a vertex v, N (v) = {U : u ∈ U if uv ∈ E(G)}. The closed neighborhood of v, denoted N [v], = N (v) ∪ {v}. The neighborhood of a subset of vertices U of V (G) is the union of the neighborhoods of the vertices of U . The degree of a vertex v written deg(v) is = |U : u ∈ U if uv ∈ E(G)|. We call N (v) the neighborhood of v and define it to be the set of vertices adjacent with v. Notice |N (v)| = deg(v). The minimum degree of G written δ(G) is the min {deg(v)}.

v∈V (G)

The maximum degree of G written ∆(G) is maxv∈V (G) {deg(v)}. 8

Definition of Graphic. These definitions, taken together with the definition of a degree sequence π(G) form the basis for almost any discussion in graph theory. Size of a graph m(G), and order of a graph, n(G) can be derived for the degree sequence: a list of the vertex degrees of the vertices in G. Any set of vertex degrees can be used to build a graph, but the same is not true for the list of vertex degrees. That is, some degree sequences are not graphic. Properties of Graphic Sequences. A property of a degree sequence is either potential or forcible. If the property P is a forcible property of degee sequence π, then every graphic realization of the degree sequence π has the property P. If the property P is a potential property of degree sequence π, then there is at least one graphic realization of the degree sequence π that has property P. Some properties, such as hamiltonicity, can be forcible or potential depending on the sequence π. In the case of hamiltonicity, an degree sequence π with smallest element greater than n/2 is forcibly hamiltonian. However, the degree sequence 2n is potentially hamiltonian and has a unique realization with a hamiltonian cycle. Graphs in general have a cycle spectrum, a list of lengths of cycles in the graph with may or may not compose the set, 3, 4, ..., n(G). In general, maximal outerplanar graphs have the full cycle spectrum: Take the longest cycle, the cycle must be hamiltonian. Then we can iteratively reduce the length of this hamiltonian cycle by 1. The clique number of a graph is the size of the largest clique in the graph. In the case of perfect graphs, the clique number, ω(G) = χ(G), where χ(G) is the chromatic number of the graph G. The chromatic number of a graph is the size of the smallest partition of the vertex sets into independent sets, that is, sets such that no two vertices in the set are incident by an edge. Sets of Graphs. The union of two graphs G and H has G ∪ H =< V (G) ∪ V (H), E(G) ∪ E(H) >. The join of two graphs G ∧ H =< V (G) ∪ V (H), E(G)∪E(H)∪E([V (H), V (G)]) >. The direct product of two graphs G×H =< {vi,j : vi ∈ V (H) ∧ vj ∈ V (G)}, {vi,j vh,k : vi vh ∈ E(H) ∨ vj vk ∈ E(G)} > . The lexicographic product G ×lex H =< {vi,j : vi ∈ V (H) ∧ vj ∈ V (G)}, {vi,j vh,k : vj vk ∈ E(G)∨(vi vh ∈ E(H)∧j = k)} > . An induced graph H is a subgraph of G such that V (H) ⊂ V (G) and E(H) = {e = uv : e ∈ E(G); u, v ∈ V (H)}. Sets of induced graphs can be used to characterize types of graphs and, in particular, a graph that has no induced subgraph H is called H-free, though this term is used to described a graph with no H-subgraph, induced or otherwise. Particularly sets of forbidden induced graphs characterize the types of graphs known as line graphs and there is a characterization of planar graphs in terms of forbidden non-induced subgraphs. In fact, extremal graph theory is often termed the study of forbidden subgraphs and an extremal graph is usually a graph that is H-free in the more general sense of the term. A complete graph is also known as clique and a complete bipartite graph is also known as a biclique. The term χ-clique could be used to describe a complete multipartite graph, but this term is not generally used. The term k-clique refers to a clique of order k. Using the above definitions, we can broaden our set of known types of graphs. The graph Ck ×Cj is called a toriod. The graph Pk × Pj is called a grid. A collection of graphs H is called an H-forest. That is, a collection of toroids is called a toriod forest. 9

This term is not always the preferred term however. For instance, a collection of cliques is generally called a union of cliques and not a clique forest. In particular, the term forest applies to trees and can be used to describe a collection of a set of trees that all share a parameter such as a path forest, also called a linear forest. Caterpillar forest is an obvious use of the forest term. A caterpillar is a tree with no subtree isomorphic to {v1 v2 , v2 v3 , v3 v4 , v4 v5 , v3 v6 , v6 v7 }. A spider is a linear forest with a vertex joined to the tail of each vertex in the path. The term join can be used to refer to inserting a set of edges from a point to a second graph. The join Γ of two graphs Γ = G ∪ H is the graph defined by < V (G) ∪ V (H), E(G) ∪ E(H) ∪ {xy : x ∈ V (G), y ∈ V (H)} > . The join of H and G is often called an (H, G)-join. The set of graphs formed by joining a vertex v to a particular graph H is called an H-cone; the vertex v is the apex of the cone. The split graph has been defined in two ways: First an independent set (tK1 ) can be joined to a clique (Kn ) to form a complete split graph where if t = n we get a balanced complete split graph. Any subgraph of this complete split graph such that we can still form a bipartition A, B of the vertex set such that the vertices in A induce a clique and vertices B still induce the complement of a clique, is a more general species of split graph. A binary tree has a number of applications in economics and is a tree that has a root with two neighbors, and each successive vertex being a leaf or having degree three. A (1, k)-tree is similar in that every vertex in the tree is of degree 1 or k. Trees are graphs of order n and size n − 1. There are other graphs of order n and size n − 1, but these graphs necessarily have cycles and have more than one component. If the (n, m)-graph G has argument m = n, and G is connected, then G is called a tree-cycle. While (n, n − 1)-graphs may or may not be acyclic, (n, n)-graphs always have a cycle and so we can view trees as extremal graphs in this cycle-avoiding sense. Collections of trees are called forests and have n(G) = m(G) + k, where k is the number of components in the forest. One of the substructures of graphs that have several applications are cycles. Cycles are uv-walks with no repeated vertices where u = v. Any closed uv-walk contains a cycle (the shortest closed sub-walk). The girth of G, generally denoted gir(G) is the length of the shortest chordless cycle in G and cir(G) is the length of the longest cycle in G. Bipartite graphs, graphs with a bipartition of the vertex set such that all edges lie between vertices in opposing vertex sets, have no cycles of odd length. A critical graph with respect to Property P is a graph such that every induced subgraph formed by removing a set of vertices from the graph does not have Property P . An edge-critical graph loses Property P if any single edge is removed from the graph. Definition 2 The graph Gk is the graph formed by adding edges uv to G whenever G has a path of length ≤ k between u and v. If G has order n, then the smallest k such that G ⊂ Pnk is the bandwidth of G. The circular bandwidth of k is the smallest absolute value of any difference between vertex labels in among the set of vertex labelings of a graph G. We will 10

continue discussing this concept later, the difference between vertex labels can be considered an edge label (edge difference). Problems. 1. Give the bandwidth of a line graph. 2. Catalog the trees of order less than 10. 3. Find the bandwidth of Pk × Pk . 4. Find the maximum power k of a graph G such that Gk is non-hamiltonian for arbitrary G. 5. Enumerate the spiders of a given order. 6. Show any two non-isomorphic trees of order n can be packed in Kn .

2.2

Digraphs

A semi-complete digraph D is a simple digraph such that Kn underlies D. If a graph has at most k vertices of the same degree then we say it is k-irregular. An orientation of a multigraph is a 1-1 onto function from the edge set of a multigraph to the arc set of a digraph with the same vertex set. ~ of a digraph D ~ is a list of the differences of ~ D) An imbalance sequence im( ~ There is a digraphic realization of an imbalance deg + (v)−deg − (v) for v ∈ V (D). ~ if and only if the condition sequence im i=k X i=1

ai ≤ k(n − k)

holds for all 1 ≤ k < n. See [60]. Theorem 2.2.1. [Ford-Fulkerson] Let ~π = {ai , bi }i=n i=1 . If for all subsets I ⊂ [n], it follows that X X X {ai , |I|}, bi ≤ {ai , |I| − 1} + I

I

[n]−I

then ~π is graphic. The following proof is based on the proof of Erdos-Gallai due to Choudum. The proof follows by induction on the number of terms in the sequence: Let the sequence ~π be given and suppose it satisfies Fulkerson’s criteria. Then let d = (a, b) from the sequence be given and remove it from the sequence; at the same time subtract 1 from the first argument of b terms of the sequence and 1 from the second argument of a terms of the sequence. Let I be a subset of π~′ , formed by the augmentation of ~π just described. We are given that the original 11

sequence satisfies Fulkerson’s criteria and thus, for all such I, I ∪ d satisfies the criteria. When we remove d, the left-hand side of the given inequality decreases by b. But the right hand side decreases by at most b as well since we augment at most b terms in the first coordinate and {ai − 1, |I| − 2} = {ai , |I| − 1} − 1. i=n Theorem 2.2.2. [Berge] Suppose that s1 ≥ s2 ≥ ... ≥ sn . The set {(ri , si )}i=1 Pj=k Pi=n composes the degree pairs of a digraph if and only if, i=1 {ri , k} ≥ j=1 sj P Pj=n for k ∈ [n] and i=n i=1 ri = j=1 sj .

2.3

Tournaments

Theorem 2.3.1. [Landau] The sequence {si }i=n is the score sequence of an i=1
Pi=n n-vertex tournament if and only if i=1 si ≥ k2 with equality when k = n. Suppose there are k scores. The k scores in question must add to more than
k . (Consider the subtournament on the k vertices represented by the scores.) 2
Suppose we have k scores that sum to ≥ k2 . Consider the n scores together so that   i=n X n si = 2 i=1 to begin with. Form the bipartite split of the tournament. Form a 1-factorization and send the scores out along the factors to realize the score sequence in some fashion. The formal definition of a tournament follows here. A tournament is an orientation of Kn . We go over the structures which are analogous to the basic structures of path, and cycle in a graph. A directed path is an orientation of a path such that only the terminal and initial vertices do not have degree (1, 1). A directed cycle is an orientation of a cycle such that all the vertices have degree (1, 1). We say an orientation of a graph is traceable if there is a spanning directed hamiltonian path. Every tournament is traceable. Theorem 2.3.2. A transitive tournament is an orientation of Kn such that the vertices have respectively degrees (n− 1, 0), (n− 2, 1), ..., (0, n− 1). Equivalently, a transitive tournament is acyclic, and has the property that if uv, vw ∈ A(Tn ) then uw ∈ A(Tn ). Theorem 2.3.3. If two vertices have the same degree in a tournament, then they lie on a triangle. Theorem 2.3.4. [deBruijn-van Aardenne-Ehrenfest] Let A be the adjacency matrix of a digraph. Let Mod be the matrix obtained from −A by replacing the ith diagonal entry by deg + (vi ). Then the co-factor of any entry in the ith row is the number of in-trees rooted at vi . An orientation of a simple graph is strong if for every vertex pair of vertics u and v in G, there is a cycle in the orientation of G containing u and v. A graph has a stong orientation if and only if it is 2-edge connected. A digraph D such that every vertex has distinct degree is called an irregular digraph. If a graph

12

has an orientation which corresponds to an irregular digraph then we say it has an irregular orientation. Theorem 2.3.5. [Meyniel] If D is a strong digraph without 2-loops and d(x) + d(y) ≥ 2n − 1 for all x, y such that xy 6∈ E(G) where G underlies D, then D is hamiltonian. Problems. 1. Catalogue the 2-regular digraphs of order 7. 2. Find all the multigraphs up to isomorphism that realize π = π(M ) = (34 ). Find all the multigraphs that realize π ′ = π(M ′ ) = (3)2 , (2)2 . Let V = {G : π(G) = π}, where G is a multigraph. Let V ′ = {G : π(G) = π ′ }, where G is a multigraph. Let H =< V ∪ V ′ , {uv : u = G, v = G − f ′ ∪ f, uv ~ : u = G − e, v = G} > . Draw the graph H. 3. Enumerate the number of 2-factors of order n. 4. Show that for all digraphs |N (S)| ≥ |

X

v∈S

deg + (v) −

X

deg − (v)|.

v∈S

5. Give an example of a di-sequence with no realization as a simple digraph whose underlying sequence has a realization as a simple graph. 6. Show that there is a 1:1 correspondence between tournaments of order n and subgraphs of the complete graph of order n. 7. Enumerate the set of labeled tournaments of order n. 8. Show that every sufficiently large order tournament T has a transitive subtournament of order 4. 9. Prove that there is a 1:1 correspondence between simple graphs G of order n and bigraphs SG of order 2n that have the following two properties: i. For all i, it is the case that viA 6∼ viB where 1 ≤ i ≤ n. ii. For all j, k, it is the case that viA ∼ vjB implies vjA ∼ viB . 10. A perfect graph is a graph such that the order of the largest clique ω(G) in G is equal to χ(G). Prove or disprove: A line graph GL is always a perfect graph. See [49]. 11. Show any graph such that at most 2 vertices have the same degree has an irregular orientation. 12. Prove or disprove: Planar 3-regular graphs G have χ(G) ≤ 3. 13

Chapter 3

Extremal graph theory The size, order approach to graphs has many developments. The maximum size of a graph that is triangle-free is not too difficult to demonstrate, using Mantel’s Theorem, but the number of triangle-free graphs of a larger order is not an easy number to compute. Constructing dense C4 -free graphs of given orders can be achieved in some cases with residue number theory. Building cages forms an interesting endeavor for most readers. Complete Bipartite Graphs. Once the Turan number of a complete bipartite graph has been calculated closely, it is possible to give an upper bound for any subgraph of this complete bipartite graph. The Turan r-sphere of order n is a graph consisting of ordered d-tuples which make up the vertex set, and edges between the d-tuples. That is, we have d = ⌈lnr n⌉ and V (rSP Hn ) = E(rSP Hn ) =

{(a1 , a2 , ..., a⌈lnr n⌉ )}, {(b1 , b2 , ..., b⌈lnr n⌉ )(c1 , c2 , ..., c⌈lnr n⌉ ) : bi 6= ci }.

Notice that the Turan r-sphere is Kt,s -free for all t, s such that {t, s} − 1 = r. To see this, notice that if we want a Kt,s it would require t vertices adjacent to the same set of s vertices. But if {t, s}−1 = r, there must be two vertices, one vertex from the t-set and one vertex from the s-set that have a common element from the ordered d-tuples which describe those two vertices. Furthermore, if n = t+s, the Turan r-sphere of order n associated with Kt,s looks like a standard Turan graph with some cliques deleted. The r-sphere is not the largest Kt,s -free graph of order t + s for all choices of t, s. Rather, for the star, the largest St,1 -free graph is Kt+1 − 2t K2 where t is even. This large difference between the number of edges in the r-sphere of order t + s and the extremal graph for Kt,s is typical, for we can always consider the clique of order t + s − 1. However, the r-sphere has     n n/r − (lnr n) 2 2 edges whenever the order of the sphere has n = rk , and t−1 = s−1 = r. In these cases, the size of the r-sphere reduces to cn2−1/(r ln r) asymptotically: Factor 14

the expression using the properties of logarithms. That is, we get [1 − lnrr2n ] as a result. Take the r power of the expressions to get approximately [1−(lnr n)/r2 ]r . Use lnr n = ln n/ln r to get approximately e−ln n/(r ln r) = n−1/(r ln r) . If we compare this to one of the best-known upper bounds on size we get cr1/r n2−1/r > n2−1/(r

ln r)

where c is a constant in r. We get c(r) > n(1−ln r)/(r ln r) . Notice that rSP Hn is not in general triangle-free, or by any means, bipartite. However, we can take the bipartite split of the graph, and get a graph that is bipartite and has approximately (asymptotically) twice as many edges. In addition, any Kt,t in this auxiliary graph corresponds to at least a Kj ∧Kt−j,t−j in the original graph. (When we split the graph, we do not add the edges that form the matching between identical vertices.) But rSP Hn excludes all graphs of this form; by construction, any two vertices that share a coordinate are independent, and there are only t − 1 coordinates. Similarly, rSP Hn excludes Kt(p) for all p. The following theorem has been attributed to a number of authors. Theorem 3.0.1. Let z(n, t) represent the maximum number edges in a Kt,t free subgraph H ⊂ Kn,n and let z ′ (t) be the number of vertices in a 2-colored copy of Kn,n , free from a monochromatic Kt,t . We have z(n, t) > cn2−1/(t−1) . Next, if cn2−1/(t−1) > n2 /2, then z ′ (t) is about 2(t−1) > n. If the graph G is not a complete graph, it is always possible to calculate an upper bound for the Ramsey number of this graph from the Turan number. In the case of Kt,t , we get the following. The Ramsey number r(Kt,t , Kt,t ) is asymptotically bounded above by the expression 2t ln t . We get this expression by comparing the size of Kn to the Turan number for Kt,t . The relevant calculation is 21 > n−1/(t ln t) , or rather, 2t ln t > n. Since the cube Q3 = K3,3 − 3K2 and in general, Qt ⊂ Kt,t , this provides a partial solution to the conjecture that r(Qn , Qn ) ≤ c2n . For small values of n, we can build a Kt,t -excluded graph from rSP Hn . Consider the K6,6 free graph on 25 vertices 3SP H25 . In blue, we get cliques of size 5 that form transversals, and the Turan-type graph 3SP H25 , in red. To see that the blue cliques do not cover a K6,6 , notice that any such monochromatic graph would have to have 2 sets of opposing vertices with coordinates in one set matching a coordinate in the other set according to the construction. But there are only 2 parameters in each coordinate set, as opposed to the 6 that would be required. Compare this to 125 < 25 ln 5 , approximately 264. Here, if we cover the the extremal graph 3SP Hn in red, it leaves an entire K25 in blue.

3.1

Potential

Swaps. Some degree sequences have only one realization; these sequences are said to be uniquely realizable. Potential Number. The potential number of a graph, σG is the minimum sum of the degrees in a graphic degree sequence π, such that every degree sequence with σ(π) = σG has a realization containing G as a subgraph.

15

Theorem 3.1.1. Any clique joined to any independent set forms a graph G with π(G) uniquely realizable. Theorem 3.1.2. Any two realizations of a degree sequence can be obtained from one another by performing a series of C4 -swaps, that is swaps of non-edges for edges in (edge, non-edge)-alternating C4 s. Theorem 3.1.3. Any graph G that is induced 2K2 -free has a uniquely realizable degree sequence π(G). Theorem 3.1.4. Any graph that is induced P3 -free has a degree sequence π(G) with a second realization up to labeling. Theorem 3.1.5. The value σC4 (n) = 3n − 1 if n is odd and 3n − 2 if n is even. If G contains a chordless path of size > 2 or cycle of length > 3, then there is, for all m, 0 ≤ m ≤ n(n − 1) a uniquely realizable degree sequence with sum degree sum m and no induced G. For all n + 1 ≤ m ≤ n(n − 1), n odd and all n + 2 ≤ m ≤ n(n − 1), n even, we have that if π is graphic, and the largest degree of π is ≤ n − 2, then π is potentially induced-P3 graphic. Problems. 1. Given that χ(G) = 3 and χ(H) = 4, is the largest G, H-free graph of order n isomorphic to the largest G-free graph of order n? 2. Suppose G is connected and has t edges, what are the extremal graphs for bandwidth? 3. Find a maximum σ uniquely realizable sequence that is Ks,t -free for all n. 4. What is the smallest order such that there is G such that π(G) has two realizations? 5. Given two 2-factors G and H such that n(G) = n(H), what is the maximal minimum number of C4 -swaps required to transform one realization of π(G) to the other? Give an extremal example. 6. Given two non-isomorphic trees with the same degree sequence what is the minimal number of C4 -swaps required to transform one tree to the other? 7. Give a lower bound for the number C4 -swaps between any two graphs G and H that have π(G) = π(H). 8. Show that σG exists for all G. 9. Find σKt for 4 ≤ t ≤ 7. (These values are known and are piecewise defined for varying values of the order of the underlying graphs G with degree sum σ.) 10. Determine whether every strong tournament has a (1, 1)-factor. Give a counterexample if such a counterexample exists. 16

3.2

Extremal graph theory

Turan’s Theorem. By analytic graph theory, we mean the results concerned with counting techniques used to prove and quantify packing problems and extremal graph theory. Analytic graph theory is one of the most well-developed subjects in mathematics today. 2 1 Theorem 3.2.1. [Weak Form of Turan’s Theorem] If m(G) > (1 − p−1 ) n2 , then G has a subgraph isomorphic to Kp . The Strong Form of Turan’s Theorem states that the extremal Turan graph for Kn is unique up to order. Suppose there
are two graphs, G1 , and G2 , that are H-free for H ⊂ Kr , where m(H) < r2 and n = n(G1 ) = n(G2 ) and H ⊂ G1 ∪ e and H ⊂ G2 ∪ e. It is not the case that m(G1 ) = m(G2 ) for all choices of H. For instance, consider the case when H = C2n+1 . Then G1 = K2n ∪ e and G2 = nK1 ∧ Kn+1 are both edge-maximal H-free graphs, but m(G1 ) 6= m(G2 ). Ramsey numbers. We begin by defining the Turan number t(n, G) to be the size of the largest G-free graph HG,n of order n. By G-free we mean G 6⊂ HG,n . Furthermore, r(G, G, ..., G) is the smallest order of Kn which, when given any edge-coloring in c colors, has a subgraph isomorphic to G in some monochromatic edge-color. Theorem 3.2.2. [Spencer] A tournament on n vertices contains a tournament of order k where k ≥ [log2 n] + 1. For all n there is a tournament with largest transitive tournament of order k ′ where k ′ ≤ [2log2 n] + 1. The development of the following definitions are not necessarily included in the subject of extremal graph theory. However, there is a great deal of evidence that both chromatic number χ(G) and the edge chromatic number χ′ (G) are closely related to the size of the largest clique in auxiliary graphs based on the adjacency matrix of a graph G. The chromatic number of a graph χ(G) is the minimum k = χ(G) such that there exists a map f : V (G) −→ [k] such that if uv ∈ E(G), then f (u) 6= f (v). The edge chromatic number of a graph χ′ (G) is the minimum k ′ = χ′ (G) such that there exists a map f ′ : E(G) −→ [k ′ ] such that if uvw = P3 ⊂ E(G), then f ′ (uv) 6= f ′ (vw). Problems. 1. Give the chromatic number of K2n − nK2 . 2. Find the number of edges that forces a Pk in G. 3. Suppose G is maximal planar of order n. What is the maximum n that excludes Pk . 4. State R(3, 3) and R(4, 4). 5. Prove the Strong Form of Turan’s Theorem. 6. The line graph of a graph G is the graph formed by GL =< V (GL ) = E(G), {u′ v ′ : u′ = e1 , v ′ = e2 ; e1 ∼ e2 } > . Prove or disprove: The edge chromatic number χ′ (G) = χ(GL ). 17

7. Prove or disprove: R(5, 5) = 43. 8. Prove the Erdos-Gallai criteria for graphic sequences. 9. Prove the Handshake Lemma: No simple graph is fully irregular. 10. Prove: If π has a realization G with H of order n having H ⊂ G, then π has a realization with H ⊂ G such that V (H) has the degG (v) ≥ degG (w) for all v ∈ V (H) and all w ∈ V (G) − V (H).

3.3

Non-planarity

Generally r is the number of regions in an embedding of G on a surface of genus k. If G is a graph embedded on a surface of genus k, then m = n − r + 2k. For maximal planar graphs, m = 3n − 6. A graph is planar if and only if it has no K5 - or K3,3 -minor. Problems. 1. Prove that the graph Kn has θ(Kn ) = ⌊ n+7 6 ⌋. See [77]. 2. Prove that the graph Kn has θ(Kn,n ) = ⌊ n+5 4 ⌋. 3. Prove that the graph Qn has θ(Kn ) = ⌈ n+1 4 ⌉.

3.4

Eulerian graphs

Eulerian circuits are spanning circuits of graphs. A walk is a list v1 , e1 , v2 , ...ek , vk of vertices and edges such that for all 1 ≤ i ≤ k, the edge ei is the unordered pair vi−1 vi . The length of a walk is the number of edges in the walk. Walks are often abbreviated by just listing the order of vertices. A trail is a walk with no repeated edge. A circuit is a closed trail; that is a circuit is a trail whose initial and terminal vertices are identical. An Eulerian circuit C is a circuit in G such that E(G) ⊂ E(C). A graph G is Eulerian if and only if the degree of every vertex in G is even and the graph is connected. It is not always trivial to build an Eulerian circuit in a graph, even once it is obvious the graph is Eulerian. However, there is a simple algorithm that achieves the construction in every case. A graph that is Eulerian has a decomposition into cycles. Theorem 3.4.1. If G has 2k vertices of odd degree, then the graph can be partitioned into k open trails each of which starts and ends at a vertex of odd degree.

18

Chapter 4

Groups and graphs The number τ (G) is the number of labeled spanning trees of G. For instance, graph τ (C6 ) = 6. But C6 × C6 has more than 36 labeled spanning trees. The labeling operation can be used to describe the type of spanning trees and subtrees in a given graph. Take a difference graph G. The set of spanning trees of G, whose edge-labels, induced by the vertex-labels fixed by the original labeling of G form a difference set, are a subset of the difference set on G. Labeling vertices also helps to count the size of the set of spanning trees in a graph. If we add one vertex to the graph, in either partite set, there are m ways that the vertex can be added in order for the vertex to be a leaf. If we add a vertex to each partite set, there are mn ways that each vertex can be a leaf. So there are m(τ ′ (Km,n )) + (n − 1)(τ ′ (Km+1,n−1 )) + mn(τ ′ (Km,n−1 )) spanning trees in Km+1,n where τ ′ (G) is the number of spanning trees up to labeled partite sets. So we get f (m + 1, n) = mf (m, n) + (n − 1)f (m + 1, n − 1) + (mn)f (m, n − 1). There are techniques for generalizing the preceding arguments as well. These techniques have been more or less outlined, explained, and demonstrated in a number of previous sources. See [64]. Some of the principles Matrix-Tree Theorem, which describes an algorithm for extracting τ (G) from the adjacency matrix A(G), are illustrated by the following facts and problems. The eigenvalues of Ak are the eigenvalues of A. For Ax = λx implies Ak x = λk x, by repeated multiplication. Using an arbitrary eigenvector ensures that the multiplicities of the eigenvalues do not change. Take the following argument. If f (x) = xT Ax where A is a real symmetric matrix, then f attains it maximum and minimum over unit vectors x at eigenvectors of A, where it equals the corresponding eigenvalues. As an aside, notice the diameter of a graph G is less than the number of distinct eigenvalues of G. (Show A0 , A, ..., Ak are linearly independent when k ≤ diam(G). It suffices to show for k ≤ diam(G) that Ak 19

is not a linear combination of A0 , A, ..., Ak−1 . Choose vi , vj ∈ V (G) such that d(vi , vj ) = k. By counting walks, we have Aki,j 6= 0, but that Ati,j = 0 for t < k. Therefore, Ak is not a linear combination of smaller powers.) If G′ is an induced subgraph of G, then λmin (G) ≤ λmin (G′ ) ≤ λmax (G′ ) ≤ λmax (G). Since A is a real symmetric matrix, λmin (G) ≤ xT Ax ≤ λmax (G) for every unit vector x. Consider the adjacency matrix A′ of G′ . By permuting the vertices of G, we can view A′ as an upper left principal submatrix of A. Let z ′ be the unit eigenvector of A′ such that A′ z ′ = λmin (A′ )z ′ . Let z be the unit vector in Rn obtained by appending zeros to z ′ . Then λmin (G′ ) = z ′T A′ z ′ = z T Az ≤ λmin (G). Now, for every graph G, χ(G) ≤ 1 + λmax (G). If χ(G) = k, then we can successively delete vertices without reducing the chromatic number until we obtain a subgraph H such that χ(H − v) = k − 1 for all v ∈ V (H). The eigenvalue of G with largest absolute value is ∆(G) if and only if some component of G is ∆(G)-regular. The multiplicity of ∆(G) as an eigenvalue is the number of ∆(G)-regular components. A graph G is regular and connected if and only if J, the matrix of all 1 entries, is a linear combination of powers of A(G). If J is of the desired form, there is a walk of length k is G between all vi , vj in V (G). Hence G is connected. To see regularity consider JA and AJ. Since J is a linear combination of powers of A, J commutes with A. Therefore, JA = AJ. Thus the (i, j)-position of A is deg(vi ) = deg(vj ). The graph G is regular. To see the necessity of the condition, notice that since G is k-regular, k is an eigenvalue of A(G). The minimal polynomial is (x − k)g(x) for some polynomial g(x). Since the characteristic polynomial of A evaluated at A is the zero operator, we have Ag(A) = kg(A). Hence each column of g(A) is an eigenvector of A with eigenvalue k. Since G is regular and connected, each such eigenvector is a multiple of the (1 × n) vector of 1 entries. Hence the columns of g(A) are constant. However, g(A) is a linear combination of powers of a symmetric matrix and therefore must itself be symmetric. Hence the columns are equal and g(A) is a multiple of J. The characteristic polynomial of Gc = (−1)n det[(−λ − 1)I − A(G) + J]. Problems. E1. If G is k-regular, then G and Gc have the same eigenvectors. E2. For each n(G), find the maximum number of edges m(G) such that G has no (1, k)-subtree. E3. Find an expression for the number of k-cycles rooted at v for each v ∈ V (G).

20

mt 2 E4. If G is k-regular and connected with spectrum k 1 , λm 2 , ..., λt , then the number of spanning trees of G,

τ (G) = n−1 Πt (k − λt )mt .

4.1

Symmetry

Graphs can have large numbers for automorphism group and be highly assymmetric in the degree sequence. Consider collections of the half-complete graph mHn . The automorphism group is 2m (m!). n! Theorem 4.1.1. The number of labelings of a graph of order n is |Aut(G)| . Cayley graphs are multi-digraph representations of groups which may have loops. Presentations are subgraphs of Cayley graphs. There are 16 groups of order ≤ 10, and thus there are 16 Cayley graphs of order ≤ 10. Symmetry of Decomposition. The complement operation, compl(Gn ) = Kn − Gn where Gn has n(Gn ) = n, can be used to transform problems like the chromatic number problem to the minimum clique cover problem. For the time being, write compl(G) = Gc . The following is known as the Nordhaus-Gaddum condition; there are other Nordhaus-Gaddum type results on other parameters, some of which are phrased in directly analogous fashion. Theorem 4.1.2. If G has order n, then i. 2n1/2 ≤ χ(G) + χ(Gc ) ≤ n + 1, and 2 ii. n ≤ χ(G)χ(Gc ) ≤ ( n+1 2 ) .

Furthermore, if G and H are placed in Kn , the resultant edge-disjoint placement has chromatic number less than or equal to the product of the chromatic numbers of G and H respectively. To continue, we will mention the half-complete graph Hn . The graph Hn and the graph Hn+1 decompose Kn+1 . While Hn is not isomorphic to Hn+1 , the complement of Hn+1 is Hn and Hn ⊂ Hn+1 . This is a very unusual set of conditions. The graph Hn is defined as Hn =< {xi : i ∈ [n]}, {xi xj f or i + j > n} > . Burnside’s Theorem. Now, k=

1 X 1 X |Gx | = |Fg |. |G| |G| x∈X

g∈G

Let G be a permutation group on a finite set X. Define x ∼ y if and only if there exists g ∈ G with g(x) = y. Then ∼ is an equivalence relation. To check reflexivity, notice g(x) = x if g is the identity. If x ∼ y then g(x) = y and so g −1 (y) = x. Therefore, y ∼ x. If x ∼ y and y ∼ z, then g1 (x) = y and g2 (y) = z and therefore g1 g2 ∈ G and g1 g2 (x) = z. Let {P1 , P2 , ..., Pk } be the equivalence classes generated by the relation ∼ . Let Gx = {f ∈ G : f (x) = x} and Fg = {x ∈ X : g(x) = x}. 21

A code C is t-error correcting if and only if d(C) ≥ 2t + 1. If C is a t-error correcting (n, b) code, then |C| ≤ bn /s where       n n n s= + (b − 1) + ... + (b − t)t . 0 1 t For instance, in a 1-error correcting (7, 2) = (n, b) code, there are 16 code words. The largest size for a (2k − 1)-error correcting (n, 2) binary code where the code words k = P (Kk , Kn ).

4.2

Clique decomposition theorem

Theorem 4.2.1. [Wilson] Let H be a graph with h
edges. There exists N = N (H) and ǫ = ǫ > 0 such that for all n > N, if h| n2 and gcd(H)|n − 1, then Kn has an H-decomposition. Block designs have been studied extensively. Generally, a block design is a set of sets of complete graphs which partition the edge set of another complete graph. The block design can be resolvable, or incomplete. If the block design is incomplete, then the set of sets does not have the characteristic that each set describes a factor of the original complete graph. While it is not always the case that a block design describes a set of complete graphs of identical order, that is typically the case. A resolvable decomposition is a decomposition of a graph that can be partitioned into a set of H-factors. Theorem 4.2.2. Any K2n has a resolvable decomposition into matchings. Theorem 4.2.3. If n is prime, then Knt has a decomposition into complete graphs of order n. Form a complete multipartite graph with n-partite sets. Use induction to decompose n complete graphs of order nt−1 . Index the vertices of the partite sets identically across all partite sets. Then use a set of n mutually orthogonal latin squares of order n to decompose the multipartite graph into smaller multipartite graphs: form blocks of vertices of order nt−2 and condense the block to form an n × n matrix. After the decomposition, expand the graph and uses induction. Theorem 4.2.4. If the size of Kn is divisible by 3, then K3 decomposes Kn . Theorem 4.2.5. Let H be a fixed nonempty graph. There exists a positive integer n0 = n0 (H), and a small positive constant γ = γ(H), such that if G is a graph with n > n0 vertices, and δ(G) ≥ (1 − γ)n, and Wilson’s conditions hold, then G has an H-decomposition. For instance, every finite family of graphs which possesses the common gcd property is totally list-decomposable. Here, the gcd(L) must divide γ(n − 1) where here gcd(L) = gcd({gcd(Hi )}) and L = {Hi }. There are the following lemmas on list-decomposability and H-decompositions. Lemma 4.2.6. [Gustavsson] Let H be a graph with h edges. There exists N = N (H) and ǫ = ǫ(H) > 0 such that for all n > N, if G is a graph on n vertices, with δ(G) ≥ n(1 − ǫ), gcd(H)|gcd(G) and h|m, then G has an Hdecomposition. 22

Chapter 5

Algorithmic graph theory The isomorphism question can be phrased in similar fashion to the decomposition question. We catalogue some types of graphs and especially emphasize that while each category can be partitioned by edge-density, some properties span the range of possible densities. One exercise is tree cataloguing and using tree catalogues to help span the set of graphic sequences of a given order. We use techniques from extremal graph theory to prove existence results in the theory of graph packing. We notice there are also labeling techniques that can give better results than greedy packings based on theorems from extremal graph theory. Also, many of the more modern labeling techniques rely on sophisticated algebra and concept of the greatest common divisor of the degrees of a graph G. Recall the notion of an adjacency matrix and notice that in some cases, linear algebra can be applied to derive complete characterizations of some graph-theoretical parameters.

5.1

Searches and labeling

Width. A tree-decomposition of a graph is a pair (T, X) where T is a tree and X = (X t : t ∈ V (T )) is a family of sets such that • the union ∪X t = V (G), • for every edge of G there is a t ∈ V (T ) such that e has both ends in X t , • whenever t, t′ , t′′ ∈ V (T ) and t′ is on the path betwen t and t′′ then ′′ ′ Xt ∩ Xt ⊂ Xt . The tree-width of a tree-decomposition of G is maxt∈V (T ) (|X t | − 1). There are many aspects to the idea of computability in regards to graph theory. The main question we are concerned with is the run-time of various deterministic algorithms. When the run-time is non-polynomial, the algorithms tend to be close to exhaustive brute-force techniques for all intents and purposes. The idea of tree-width was implemented to fragment graphs into classes that have better 23

run-time according to the how tree-like the edge set appears in terms of a minor that covers the vertex set. It is known that for graphs of bounded tree-width, NP-complete and NP-hard problems often have polynomial-time algorithms. Examples include independent set, chromatic number, clique number, and some restricted coloring problems that are also essentially chromatic number problems. Notice that the definition of tree-width has been used to study the problem of whether or not every surface has a finite set of excluded minors. It has been conjectured that the cardinality of such a set is 1. Vertex-Labeling. Much of the study of vertex-labelings is motivated by the following conjecture. Conjecture 5.1.1. [Ringel] If Tk is a tree of order k, then Tk |K2k−1 . The respective names on the vertices of a graph are arbitrary, but they are generally denoted {v1 , v2 , ..., vn } where n = |V (G)|. However, special properties of the graph can often be discerned by permuting the vertex indices in some fashion. For instance, for a balanced bipartite graph, we might index all the vertices in one partite set with even integers. We may even call this indexing of the vertices a vertex-labeling, and we introduce the following definition to clarify this notion. A proper-vertex labeling is defined in the following way. A proper vertex-labeling of G is a bijection from V (G) to [|V (G)|]. Similar is the notion of a proper-edge labeling, though the applications of this second notion are less intuitive in most cases. A proper edge-labeling of G is a bijection from E(G) to [|E(G)|]. We will discuss some of the examples of proper-edge labelings later and include some examples of improper edge-labelings at the same time. We will see the following definitions are particularly useful when discussing graph-labeling problems. If f is a proper vertex-labeling of G, then any gf defined by gf (e) = g(f (u), f (v)) is called the edge-labeling of G induced by f. In general, notice that we can define gf without defining f first so that gf (e) has various values for various values of f. In general, we want the situation where gf is defined so that there exists a proper-vertex labeling f of G that induces a proper edge-labeling gf of G. Much of the subject of graph-labeling can be understood as an attempt to find general conditions for (f, gf ) to be a proper (vertex-labeling, edge-labeling) pair for more and more general sets of graphs. Let Gn be a caterpillar of order n. Let S be the ordered set {0, 1, ..., n − 1}. Let P be the ordered set S − U. Let U be the ordered set {0, 1, ..., k, n − j, ..., n − 1}. Label the head of Gn with n − 1 and set it to x. If x is the tail stop. Otherwise, label all the neighbors of f (x) with labels starting from the leftmost side of P and moving right until we label lastly the unlabeled neighbor of x in the spine of Gn , set this vertex to y. If y is the tail stop. Otherwise, label the neighbors of y with labels starting from the rightmost side of P and moving left until we label lastly the unlabeled neighbor of y in the spine of Gn , set this vertex to x. Iterate the algorithm. The next algorithm can be difficult to apply in practice due to the switching of the placement of the labels when the matchings are internal to the tree. Find a maximum matching. Remove all pendant vertices not in the matching. Contract the matching. Repeat steps until there is only a vertex left. Add back the first 24

edge and label the edge gracefully. Expand the last matching and double every vertex label in the pre-existing matching so that if a matching is internal to the tree, the fixed labels that were incident the internal vertex rotate away to (or back from) the opposite end of the matching. Fix an odd label opposite the even labels along the matching so that the labels add to n − 1 where n is the order of the new tree. Insert any leaves incident the even labels. Use the smallest difference larger than every difference incident the vertex to achieve vertex label k and then increase the label of every vertex-label ≥ k by 1. (It may be necessary to subtract every label from the order of the tree, and replace the labels by the remainder from the subtraction operation in order to apply this step correctly. The vertex labels can be flipped back afterwards.) The following theorem gives a necessary and sufficient condition for decomposition of a tree, where H|G and H is a path, star, or double star. The condition is necessary for any tree-decomposition of a tree. The vertex v is a cut-point unless it is pendant. Let u1 , u2 , ..., uℓ be the set of cut-points of a tree. The matrix Dk (T ) = dij , i = 1, 2, ..., k − 1; j = 1, 2, ..., ℓ where dij = di,k (uj ) where di,k (uj ) = deg (uj ). Theorem 5.1.2. If H is a tree with k edges and G is also a tree, then H|G only if there is a matrix X with non-negative integer entries satisfying Dk (H)X = Dk (G). Edge-Labelings. There are many forms of edge-labeling questions. We will focus on irregular labelings and irregular labelings of minimal strength. The idea behind irregular labelings is to find a multigraph MG that covers a given graph G such that there is a subgraph of the super-graph MG which covers G and has an irregular degree sequence, that it, a degree sequence π such that the multiplicity of every degree in the list is 1. Specifically an irregular labeling of a graph G is a not necessarily proper edge-labeling of the edge set of G such that X X f (uw) f (uv) = uv∈E(G)

uw∈E(G)

if and only if v = w. All connected graphs except K2 and K1 have irregular labelings. It is known that no non-trivial graphs have irregularity strength 1 and that the only connected graph with s(G) ≥ n is the triangle. Irregular labelings can be demonstrated for some quite general classes of graphs and these schemes are often very similar to the schemes for graceful labelings and antimagic labelings. The topic of irregularity strength and irregular orientations was preceded by the study of highly irregular graphs and digraphs. A graph or digraph is highly irregular if no two vertices in the same neighborhood have the same degree. A highly irregular graph H with maximum degree d has at leat 2d vertices. For every positive integer n 6= 3, 5, 7 there exists a highly irregular graph of order n. The size of a highly irregular graph of order n is at most n(n + 2)/8, and every graph of order n ≥ 2 is an induced subgraph of a highly irregular graph of order 4n − 4. Every digraph of order n ≥ 2 is an induced subdigraph of a highly irregular digraph of order 4n − 4, and 25

every finite group is isomorphic to the automorphism group of a highly irregular digraph. Define irregularity strength to be the minimum maximum edge label required in an edge-labeling so that the induced vertex-labels (edge-label sums) are all distinct is called the irregularity strength of a graph. Every graph has irregularity strength at least two and there are many papers that determine the irregularity strength of a graph class for all orders. Among these classes are suitably dense graphs, grids, toriods, paths, cycles, and some classes of trees. The irregularity strength of a graph is linked closely to a lower bound λ. The parameter λ = maxS ⌈ deg(v)−deg(w) ⌉ where S ⊂ V (G) and v is the vertex of |S| maximum degree in S, while w is the vertex of maximum degree in S. Consider an arc weighting of a digraph D, f : A(D) → Z + . Define a vertex labeling induced by our arc weighting to be g : V (D) → Z + ∪ {0} × Z + ∪ {0}, where X X f (xv)). f (vx), g(v) = ( xv∈A(D)

vx∈A(D)

If the arc labeling such that every vertex label is distinct, then the labeling is irregular. Let I(D) denote the set of irregular labelings of a digraph D. The irregularity strength of a digraph D is defined as ~s(D) = minf ∈I(D) maxe∈A(D) f (e). Again, ~s(D) is the irregularity strength of D and an irregular labeling of D with s arc labels available is an irregular s-labeling. Furthermore, we may even say an irregular s-weighting of the digraph, referring to the weighted vertex degree pairs if the context is perfectly clear and we want to emphasize some point about the vertex weights. Notice we cannot have an irregular (~s − 1)-labeling. There are quite a few known results concerning irregularity strength of digraphs. Transitive tournaments have irregularity strength 1. The strength of some orientations of some classes of graphs have been determined for all orders. The following parameter is very helpful when dealing with digraph irregularity strength. Let D be a digraph and let U ⊆ V (D) be such that for all x in U , i1 ≤ d+ (x) ≤ i2 and j1 ≤ d− (x) ≤ j2 . Then ~s(D) ≥ max

U⊂V (D)

{s : qU (s) = 0}

where qU (s) = (si2 − i1 + 1)(sj2 − j1 + 1) − |U |. Let D be a digraph with irregularity strength s. Then the degree of every vertex x in U must have i1 ≤ d+ (x) ≤ i2 and j1 ≤ d− (x) ≤ j2 . It follows that every vertex in U must have its weighted out-degree (in-degree) between i1 and si2 (j1 and sj2 ). We necessarily have that for all U ⊆ V (D), (si2 − i1 + 1)(sj2 − j1 + 1) ≥ |U |. Because this is the case for all such subsets U ⊆ V (D), the theorem follows as stated above. Define ~λ to be the largest zero of q|U| for a given graph. Let an arbitrary digraph D be given and let v ∈ V (D) be an arbitrary vertex in the vertex set of D. Then we know that Y Pv < 1 ⇒ ~s(D) ≤ s. v∈V (D)

For all digraphs D, it follows that ~s(D) = ~λ(D). 26

Notice that for undirected graphs, it is not always the case that s(G) = λ(G). If the graph G has degree sequence π(G) and X sd < 1, d2 d∈π

then s(G) = 2. The s∗ -strength of a graph s∗ (G) is the minimum irregularity strength over all orientations of G. Recall that the definition of ~λ(D) = maxU {x : qU (x) = 0} where qU (x) = (x∆+ (U ) − δ + (U ) + 1)(x∆− (U ) − δ − (U ) + 1) − |U |. It is clear that max Pv ≤

~ v∈V (D)

nk − 1 (ck s − ck + 1)2

where ~λ = max{s : (ck s − ck + 1)2 = nk } by a two case argument. Either the weightings of any two vertices in the set U are independent or they form a subset of U that generates a smaller λ. For the time being, we write λ instead of ~λ. The probability that the two vertices have the same weighting if we remove √ the arc between them is at most 1/ λ. But if we replace the arc, we get an additional factor of λ1 . So then, if ~s ≥ s it is clear that max Pv ≤

v∈V (D)

nk − |N (v)| − 1 |N (v)| < 1. + λ λ3/2 − 1

Some graphs have irregular orientations. For instance, Kn,n+1 is the only form of a complete bipartite graph with an irregular orientation. The obvious necessary condition for a graph to be irregular, that there exist ≤ k + 1 vertices of degree k is not sufficient. A k-irregular graph is a graph that has at most k vertices of any degree and k is a fixed constant. It is known that 2-irregular graphs have irregular orientations. An interesting question is to determine the minimum size of a graph with an irregular orientation for a given order. Graphs of the form Kn,n+1 and Kn can be combined in ascending size to achieve graphs that are close to the minimum size for given orders. The minimum size of an irregular digraph that allows 2-loops has been completely determined. There are many ways to approach the topic of digraph regularization and most center on adding the minimum number of vertices and then the minimum number of arcs between these vertices and the original digraph, never allowing multiple arcs. It is not clear what categories of graphs can be regularized with so-called magic labelings for digraphs or simply allowing arc multiplicities to be increased. There is some work along these lines, but most has been concerned with magic labelings of non-oriented graphs. There are digraphs which cannot be regularized without the addition of new vertices. Antimagic labelings are proper edge-labelings such that the edge-sum at every vertex is distinct. We can show a connected graph other than K2 is antimagic by finding an antimagic spanning tree and then adding back the edges incident the largest leaf vertex-label first, then the second largest leaf vertex-label and 27

so on, each time adding one to every edge-label and using the 1 edge-label on the new edge, eventually adding back all the edges. Once we cover the pendant vertices with edges, it does not matter what order we add the edges back. This follows because the edge difference we add at each stage, 1, is smaller than the difference between any two vertex-labels, which necessarily all differ by at least two, as we will see from the following algorithm. To label a tree according to the conditions of antimagic labelings, follow the algorithm below. Pick an edge in the tree that is not pendant. Label the branches of the tree. That is, the sets of vertices that pass through the same path directed to the central edge. Place the edges of the tree in classes 0,1,2,... according the the distance to a pendant vertex. Radially label the edges 0,1,2,... starting with the leaves and always labeling the leaves incident one another with consecutive labels. Proceed to the edges in class 1,2,3,... according to the following rule: place the next label incident the smallest label incident a non-labeled edge.

5.2

Decompositions

Theorem 5.2.1. Every outerplanar graph has an ascending subgraph decomposition. We can define a recursive algorithm that builds an ascending subgraph decomposition of an outerplanar graph of appropriate size. Given the union G of two ∗ ∗ edge-disjoint star-matchings H = Hn,t and H ∗ = Hn−1,t ∗ where t > t , we can ′ ′′ ′′ decompose G into H = Hn,t∗ +1 and a second star matching H = Hn−1,t ′′ ′′ ′′ where t = t or t = t − 1. Either, (i): H has a free K2 , (ii): the central node of H is not covered and one of the leaves of H is uncovered, or (iii): there is no free K2 of H and a K2 of H ∗ covers the central node of the star in H. In the first case, take H ∗ ∪ K2 = H ′ . In the second case, add the K2 of H that is uncovered to the star of H ∗ . In the third case, we get a contradiction by counting since ∆(H) ≤ ∆(H ∗ ), or else we can let H = H ′ and H ′′ = H ∗ immediately. Suppose G has a star-matching. Then G has a star-matching H of minimum degree δ = ∆(H) and if G − H has an ascending subgraph decomposition with root H ∗ , then ∆(H ∗ ) ≥ ∆(H). Discard all alternating paths and cycles in the union of H and H ∗ . We are necessarily left with both stars leaves saturated by alternating paths that lead to other leaves; there could be a path from a leaf of one star to the other star. This necessarily leaves a K2 of H ∗ free, or else the K2 is incident the center of H. In the former case, let H ′ = H ∪ K2 and we contradict the definition of H. In the latter case, swap an alternating cycle incident two of the leaves of H and add the K2 to the star on H. Since ∆(H) > ∆(H ∗ ) there must be such a cycle and we get
a contradiction with the definition of H. Every outerplanar graph of size n+1 has an Hn,t -rooted 2 ascending subgraph decomposition
for some 0 ≤ t ≤ n. Let G be a given simple outerplanar graph of size n+1 G that maximizes t. 2 . Select the Hn,t from
n Remove the graph from G. Because G − Hn,t has 2 edges it has a Hn−1,t′ rooted ascending subgraph decomposition. Now we will show we can exchange 28

edges with Hn,t and the components of our ascending subgraph decomposition of G − Hn,t to get a rooted star-matching ascending subgraph decomposition of G. For each step we will increment j by one and move to the component of size n − j. Use the remainder of Hn,t each time we iterate j. When we are done, we have an ascending subgraph decomposition of G. The following two theorems have simple proofs based on optimization and can be used to prove the ascending subgraph decomposition conjecture. The term 2-star-matching refers to a matching taken with an independent graph formed by the induced graph of two vertices H = [N [v] ∪ N [w]].
Theorem 5.2.2. If the diameter of a graph is 2, and G has n2 < m(G) ≤
n+1 2 , then G has a 2-star-matching of size n.
Theorem 5.2.3. If the diameter of a graph is > 2, and G has n2 < m(G) ≤
n+1 2 , then G has a 2-star-matching of size n. Problems. 1. Show every tree can be labeled in n(T ) labels so that no two non-isomorphic trees generate the same permutation. 2. Give an example of a disconnected graph that is antimagic. 3. Give the irregular labeling of minimum strength for paths and cycles. 4. Show caterpillars have harmonious labelings. 5. Prove Theorem 5.2.2. 6. Prove Theorem 5.2.3. 7. Prove the ascending subgraph decomposition conjecture for girth 4 graphs. 8. Prove the ascending subgraph decomposition criteria hold for a pair of oriented stars and any collection of in-stars and out-stars. Prove the ascending subgraph decomposition criteria for any transitively directed bigraph. 9. Prove the ascending subgraph decomposition conjecture. 10. Prove: If

X sd ) < 1, .1664223914...( d4 d∈π

∗

then s (G) = 2. Every d-irregular graph G has s∗ (G) = 2. 11. See [49]. Prove that if G is a unimodal caterpillar, then {Gi }i=n−1 packs in i=1 Kn . That is, show there is a set of injective vertex labelings {fi }i=n−1 such i=1 that the domain and range of each fi is restricted by fi (V (Gi )) −→ [n], and for all e, e′ we have gf (e) 6= gf (e′ ) for any e ∈ E(Gi ), e′ ∈ E(Gk ), where k 6= i. 29

Chapter 6

Probabilistic graph theory 6.1

Graph placements

There are 2m(G2 ) ways G1 can intersect G2 and this fixes the first and second choice of label for the graph G1 . That is, there are n2 (n − 1)2 [n − 2] · · · (n − n(G2 )) packings, but only 2m(G)(n) · · · (n − n(G2 )) of those packings have an intersection between G1 and G2 . Therefore, the probability G1 and G2 intersect is Pe , as claimed. Now by the definition of expectation we get that E(Xe ) = Pe if Xe is the number of edges of Pin a packing of G1 and G2 . Then Pintersection the lemma follows because E( ei Xei ) = ei E(Xei ) where {eP i } is the set of edges of G1 , by the linearity of expectation. We know that E( ei Xei ) is the expected number of edge duplications of G1 with G2 because the number of intersections of G1 is the sum of the intersections of its edges. The following theorem is the overall goal. We delay the proof until we can build some familiarity with the tools necessary to prove the result. If T1 , ..., Tn is a list of trees indexed by size, then the list of graphs decompose Kn+1 . We write σ({Li }) to denote the number of edges left uncovered by a set of injective maps fi (V (Li )) → V (Kn ), where here Kn is implicitly meant to be a labeled copy of the complete graph of order n where the vertices are not interchangeable up to indexing. The vertex maps are injective and if we write σ({Li }) they are understood to be fixed. The set of vertex maps and the induced maps on the edge sets of the Li are called a packing of the list of graphs and are also said to be a placement of the elements of the list. Generally, we place a list in Kn , but there are also situations which call for the placement of a set of graphs in some other graph. The range space of the vertex maps and the range space of the induced edge maps is a subspace of a graph G that is often referred to as a super-graph of the range space. If we place {Li } in a graph other than Kn , we write σG ({Li }) to mean the number of edges left uncovered in G. Here, the functions that map Li into the edge set of G should be given explicitly. The value E(σ({Li })) is defined in the obvious way and is the average number of edges left uncovered as the maps {fi } on the {V (Li )} range from all possible 30

injective maps into [n] injectively. If E[σ({Li })|σ({Li }) ≥ 1] > E[σ({Li })], then there is a packing of the list {Li } in Kn . If σ({Li }) 6= 0, then σ({Li }) is called the slack of the placement. If σ({Li }) = 0, then the placement is said to be dense. A set of maps on a list of graphs into a super-graph that does not map any two edges from the list of graphs onto the same edge of the supergraph is said to be an edge-disjoint placement. A dense placement of a list of graphs that is also an edge-disjoint placement of the list is a decomposition of the super-graph. Conjecture 6.1.1. [Bollobas-Eldridge] If n ≥ 6, (∆(G)+ 1)(∆(H)+ 1) ≤ n+ 1, n(G) ≤ n, and n(H) ≤ n, then there is an edge-disjoint placement of G and H in Kn . We have that if n(G), n(H) ≤ n, we want that m(G)m(H) m(G0 )m(H0 )

≥ + E(X) + 1 n n−2 2

2

where here E(X) is the expected number of edge-overlaps at an edge in the = double star rooted at the given edge-overlap. Use the equalities ∆(G)n 2 m(G) and ∆(H)n = m(H); that is, assume G and H are regular in order 2

to simplify the
calculations. Multiply the inequality by n2 n−2 2 , move the m(G0 )m(H0 ) n2 term to the left-hand side, and re-group the E(X) + 1 term to
get (∆(G)∆(H))n(n − 1)(n − 3). Now divide both sides of the inequality by n2 . Again, use the assumption that G and H are regular and bound the expression on the left-hand side in the following way: 2(n + 2)(∆(G)∆(H)) −

(n − 4) (4n − 6) (n(∆(G)∆(H)) − (∆(G) + ∆(H)) + 1 4(n − 1) 2

(n − 4) (∆(G) + ∆(H)) + 1. 2 The second step is a slight bound of the left-hand side. We increase the left-hand side to make the inequality tighter. Now it is easy to see that the inequality holds in all cases where the hypothesis holds. The term 2n(∆(G))(∆(H)) is larger if G and H are regular, which makes the inequality tighter and is the only term affected by assuming G and H to be regular. The following proof relies on the tools we have used in the previous two sections. We introduce a definition for the purpose of the proof. A star-resolution of a tree Tn is the shortest possible list of trees ≤ (n + 5)(∆(G)∆(H)) −

Sn = A0 , A1 , ..., Ak = Tn where m(Sn ) = m(Tn ) = m(At ) for 0 ≤ t ≤ k, and where At+1 is formed by replacing an edge ex = xz, where x = ∆(At ) is fixed throughout, with the edge yz. Notice that if {Li } is a list of paths that pack in Kn it does not necessarily hold that {L′i } packs in Kn where {L′i } is a list of stars and where m(Li ) = m(L′i ), 31

the obvious counterexample being the hamiltonian path decomposition of Kn for n even. Consider that E(σ({Si })|σ({Si }) ≥ 1) > E(σ({Si })) under the condition that the Li are formed from the Si element-wise by starresolutions. The proof is by induction on the largest i we resolve and the greatest k in the star-resolution. Consider that if we have an edge-disjoint placement of the list {Li } − Lj (= At ) in Kn and we take the expectation of the number of edge-overlaps in the various placements of Lj in the packing, we get that there is a greater conditional expectation once there is a first edge-overlap by induction. Now let L′j be another tree in the chain of the star-resolution, At+1 . By taking the packings of Lj in the placement of the {Li } − Lj and replacing the edge xz in the manner of the star-resolution, the expected value stays fixed or increases, as long as there is an edge-duplication in both trees. Case 1. If e′ = e = uv is the only duplicated edge in the packing, then by assumption moving the edge reproduces the duplication somewhere else. Case 2. Suppose the edge ej ∈ Lj − L′j has ej ∼ e′ (= u′ v ′ ) ∈ L′j . Then observe that given two stars S1,m1 and S1,m2 which are subgraphs of S1,m rooted at the vertex v, the expected value E(X ′ ) of the edge-overlap of the two sub-stars has E(X ′ ) =

m1 m2 m

if the edges of the two sub-stars are distributed uniformly in the larger superstar. Count the packings of e′ overlapping e(= uv) ∈ Li where u → u′ and v → v ′ together as the same total expected change in slack and observe this expectation is 0. Case 3. We will show in all remaining cases that if L′j intersects some Li ∈ {Li } − Lj , at the edge e, then the average slack increases from the expected values over all packings. Consider the inequality (1) 2m(H) ≤ dLi (e). (n − 3) Since the conditional expecation of slack was greater than the expectation of the slack for the set {Li }. Consider the case of {L′i } where Lj → L′j . By double induction on (i, t), the result follows. Problems. 1. Characterize the sets of stars that decompose Kn . 2. Show almost all graphs have δ(G) > n/2. 3. Show almost all graphs are connected.

32

Chapter 7

Surfaces 7.1

Introduction

We suppose the reader is familiar with the terms surrounding the concept of chromatic number and the definition of a minor. A proper vertex coloring is an injection of V (G) → [k] such that if v ∈ V (G) and w ∈ V (G) are mapped to the same integer i, then v 6∼ w (vw 6∈ E(G)). Vertices in the same color class necessarily form an independent set. The parameter χ(G) is the least k such that such an injection from V (G) to [k] exists, and is therefore the smallest number of equivalence classes needed to partition the V (G) according to the relation 6∼. Here, we say k-chromatic to mean G has χ(G) = k. Notationally, we have that if a vertex v is colored color j in a proper coloring f of G, then we write col(v) = j. It is understood that col(v) = colf (v). Every k-chromatic graph has a Kk -minor. Stating the definition of a minor seems to necessitate stating the definition of an edge contraction. An edge contraction in a graph G is an operation on the vertex set V (G) and the edge set E(G) of a graph G wherein for some edge uv = e ∈ E(G) we replace u, v ∈ V (G) with a vertex v ′ and N (u) ∪ N (v) with N (v ′ ) = {v ′ x : x ∈ (N (v) − u) ∪ (N (u) − v)} producing H. If vx and ux are both in E(G) then we replace these edges by a single edge v ′ x. An H-minor of a graph G is a subgraph of G that can be contracted to H by applying the graph operations of vertex and edge deletion and the operation of edge contraction. Finally, we will need the definition of an induced subgraph in G and define [V0 ] to be the subgraph of G induced by the set V0 ⊂ V (G).

7.2

Color-alternating complexes

A path P ⊂ G colored in two colors in some properly colored graph G is called a color-alternating path. Such a path P = v1 v2 ...vn has a color-alternating property such that col(vi ) 6= col(vi+1 ) for 1 ≤ i ≤ n − 1, but col(vi ) = col(vi+2 ) for 1 ≤ i ≤ n − 2. A color-alternating complex ofSorder k or just k-complex,
Mk , is the union of k2 color-alternating paths Pvi ,vj where vi 6= vj and 33

1 ≤ i, j ≤ k are a set of k vertices {vi }i=k of the i=1 which form the endpoints
color-alternating paths. That is, the complex is the union of k2 paths and for every path there are two distinct points vi and vj from the set {vi } which are colored distinctly and form the endpoints of the color-alternating path Pvi ,vj while col(vi ) and col(vj ) form the colors on the color-alternating path Pvi ,vj . We say that a k-complex is (based) on the set V if V = {vi }i=k i=1 and equivalently, in this situation, that the set V forms a basis for the k-complex. A uniquely colorable graph has only one partition of the vertex set into independent sets (up to ordering) such that the cardinality of the partition is χ(G). Let v be the final vertex of the basis of the k-complex and index the paths of the Mk−1 complex formed by the other k − 1 vertices which intersect the Pv,vi : 1,2,...,t. Notice the path Pv,vi can only intersect paths of the form Pvi ,vj . Contract the remaining paths of the Mk−1 complex to edges according to inductive hypothesis. Now, on the remaining paths, contract any edge with an endpoint of degree 2. Next, proceed through the indices of the paths deleting any edges between the image of path t′ and the images of paths of lower index t′′ unless these paths share an argument. That is, do not delete edges between paths of the form Pv,vi and Pv,vj on Pvi ,vj , but delete edges between Pv,vi and Pw,vj on Pvi ,vj if v 6= w. Now edge-contract each path Pw,vi to w starting with the path of lowest index until we reach a vertex x of degree 3 or more such that wx = e ∈ E(G′ ) where G′ is our edge-contracted and edge-deleted graph. If this vertex is the intersection of the image of one or more Pw,vi paths contract x to w. If this vertex is the intersection of the image of one or more Pvj ,vi paths, contract x to vi along whichever path-image in the intersection still connects x to vi . It follows from our construction that at least one of these paths still connects x to vi . For when we edge-delete between detours P3 = D(P1 , P2 ) and P2 = D(P1 , P3 ) where P1 = Pvj ,vi and P2 = Pvh ,vi we leave P1 and follow P2 to vi . Since there is always a path from vj to vi as we proceed through the indices of our paths deleting edges, and no deletion destroys all such paths, it follows that when we are done there is still one such path. Furthermore, as we contract x to vi it follows from our construction we do not separate the image of any path whose endpoints are both distinct from vi ; such a path would have been a detour from the path we are following and we would have deleted the edge between these two paths. Finally, any path with one endpoint as vi still has a path to its other endpoint under this operation since we never disconnected the two endpoints in the initial edge deletion process.

7.3

Planar graphs

Suppose that there is a maximally planar graph that requires 5-colors. Then this graph has a K5 -minor. However, then, this graph is non-planar by Kuratowski’s Theorem. The reader is referred to [49] for the more basic definitions and terms. The Four Color Theorem was first proved by the introduction of obstruction sets. By showing that a graph must be outside an obstruction set to be a counterex-

34

ample to the Four Color Theorem, and that a member of this same obstruction set must be included in any graph that is a counterexample, it has been demonstrated that the Four Color Theorem holds. In both the Appel-HakenKoch and Robertson-Sanders-Seymour-Thomas proofs, the key approach was to prove, using a set of discharging rules, that any maximal planar graph that has a particular set of configurations as a subgraph could be colored with just four colors, and a proof, using reducibility criterion, that the set was unavoidable in any counterexample to the Four Color Theorem. The method of discharging has been used in several other problems [53], [49]. Here, we attempt to describe the Robertson, Sanders, Seymour, and Thomas proof in a very brief manner. The set of unavoidable configurations, set A, mentioned below, is pictured in the appendix of [59]. A planar graph G is called a minimal counterexample to the Four Color Theorem if it is not 4-colorable and every planar graph G′ with |V (G′ )| + |E(G′ )| < |V (G)| + |E(G)| is 4-colorable. We say that G is internally 6-connected if it is 5-connected and for every set U ⊂ V (G) of size 5, G − U is either connected or consists of two connected components, one of which is just a vertex [53]. The neighborhood of a graph, N (G), is the union of the neighborhoods of all its vertices less the vertices in the graph. Given a cycle inscribed in the plane, the interior or exterior neighborhood of a cycle are the vertices in the neighborhood of that cycle that are all in the same region among the two regions created by removing the cycle from the plane. The (exterior) neighborhood of a tree is just the neighborhood of the tree. The coBetti number of a graph is n − m. The Betti number of a graph is n − m + k, where k is the number of components of the graph. A tree-cycle is a graph G ∪ H where G is a forest and H is a cycle whose vertex set includes exactly one vertex from every tree in G. That is, they are graphs with coBetti number 0 and Betti number 1. An even cycle is a bipartite cycle, an even tree-cycle is a bipartite tree-cycle. An outerplanar graph is a planar graph with no interior vertex; that is every vertex lies on the exterior face of an imbedding of the outerplanar graph in the plane. A two-tree factor in a graph is a pair of induced trees which are vertexdisjoint and span the vertex set of the graph. Notice the following theorems that reduce the Four Color Theorem to a theorem about hamiltonian maximal planar graphs. Any maximal planar graph is 3-connected. If a maximal planar graph is 3-connected, but not 4-connected, it has a nonfacial triangle. A two-tree factor in the dual does not appear to provide a simple algorithm for 3-coloring the edges of the dual. Furthermore, hamiltonicity in a maximal triangulated planar graph is not equivalent to a hamiltonian cycle in the dual, though there is no known counterexample to the forward direction of the implication. Also, there does not seem to be a simple way of orienting the edges of hamiltonian maximal planar graphs so that there is no directed path of order five or more, a situation complicated because the interior of the hamiltonian cycle need not be acyclic, as it may appear at first inspection (outerplanar graphs can have several endblocks). If a maximal planar graph is 4-connected, it is hamiltonian. It is conjectured that the edges of a maximally planar graph be 3-colored, so 35

that no two edges of any triangle receive the same color. A 3-edge coloring of the dual would correspond to such a coloring since edges correspond to edges under the dual operation. An orientation of a face is a labeling of the face 1 (clockwise) or 2 (counterclockwise.) A proper orientation of a maximally planar graph G is an orientation of all the faces so that all the vertices v ∈ G have face sum congruent to 0 modulo 3, where face sum is the sum of all the orientations of the faces incident v. Charge at a vertex is a subset of the face sum. Generally, it refers to face sum in an incomplete orientation of the faces of a graph G. That is, the vertex sum before all the face-labels have been filled in. An maximally planar graph G has a 4-coloring whenever it has a proper orientation. The face orientations describe a color rotation of the edge-colors at each triangular face. If the orientation is well-defined (all the face sums are 0 modulo 3), then we can pick an initial color for one edge and then the other colors of all the triangles are fixed by the rotation scheme or rather, the proper orientation. A cycle-factor of the regions of a maximally planar graph G is the set of sets of regions induced by a cycle-factor of the dual of G. A cycle of regions is one of the sets of regions that correspond to a cycle in the dual and necessarily consists of a set of regions which are linked edge to edge, so that the remaining edges of the regions form 2 cycles; all of these edges correspond to a subset of a matching in the dual. A canonical orientation of a cycle of regions is described in the following way: if two incident triangles have an extra edge along the same edge-cycle bordering the cycle of regions, then the two triangles receive different labels. If the two triangles have, respectively, their extra edges along opposing cycles enclosing the cycle of regions, then the two triangles receive the same label. The first label selected in any labeling or orientation of the cycle of regions is arbitrary. If a cycle of regions is labeled canonically, the charge, or face-sum, contributed by the region of cycles is 0 at every vertex on the cycle of regions. The only issue is whether the two vertices incident the triangle we begin with have face sum zero contributed by the cycle of regions. To see this, notice there must be a pair of triangles with their base on the same side: pick one of these triangles to begin with and then rotate away from the other triangle in the pair, reaching it second to last, arriving in a cycle after 2n steps at the first triangle which we orient with a 1. Now, there are an even number of regions and so the number of pairs of triangles with their bases switched is the same as the number of consecutive pairs triangles with identical base parity-wise, if we count all the way to the beginning triangle. Then the number of base switches must be even, so that the number of consecutive pairs are also even. That is, the second to the last triangle, the last triangle next to our initial triangle is labeled with a 2. Therefore, we can glue the two base points together and get a 0 face sum. Now then, all except the face sums at the top of the altitudes of the two triangles are 0 modulo 3. But the sum at each cycle must be the same, so that the last vertex has face sum 0 as well. To see we need only consider this case, notice that if the bases alternate throughout, there is necessarily an obvious labeling: 1 on every triangle. If every cycle of regions is labeled canonically, then the maximally planar graph is labeled properly or rather has a proper orientation 36

given the canonical labeling. The sum contributed by each cycle region is 0 at each vertex and every vertex faces a finite number of cycle regions, therefore, the sum at each vertex is 0. Pick a diamond incident v, N = {v, w, x, z; vw, wx, vw, wz, vz} and replace the edge vw with the edge xz to form the diamond N ′ = {v, w, x, z; xz, wx, vx, wz, vz}.

Let G′ = G − E(N ) ∪ E(N ′ ) = G − vw ∪ xz. G′ has a 4-coloring since degG′ v = 4. Now then by Tait’s an even 2-factor exists in the dual of G′ . Thus, by previous theorem, there exists a canonical orientation: simply pick each even cycle of regions and perform a canonical orientation. Now suppose without loss of generality that the two regions of N ′ have the same orientation under this algorithm: 1. Then we can replace the diamond N, relabel the regions 2 and we necessarily have a proper orientation of G. That is, we can remove xz and replace vw, the colors of v and w are necessarily different. This is the case in Cases 1.1-1.3. Now suppose the two regions have different orientations and we are in Cases 2.1-2.3. It is not clear immediately that these cases can be resolved. We might, as we induct, find that the diamond N ′ uses only three colors on its vertex set. Therefore, we cannot recover the diamond N, and use the proper 4-coloring of the underlying vertex set of G′ to 4-color the vertices of G by indentifying the vertices with the exact same colors in G as in G′ . In this case, the edge vw makes v and w adjacent, then we have col(v) = col(w) in G′ . To summarize what is known about the hierarchy of chromatic numbers for sparse graphs, notice the trees, outerplanar, and planar graphs form a pyramid where each step gives a another increase of one in chromatic number. There is no well-known corresponding partition of the 5,6, and 7 chromatic number graphs. Here, we investigate 2-factors on high level surfaces with the idea that this question may shed some light on the chromatic number question. An (n, m, β)graph is a graph with order n, size m, and coBetti number β. We propose the following theorem. Any triangulation of a surface with a spanning triangle-free (n, m − r, 2 − 2g)-graph has a 2-factor in its dual. Since the factor is spanning and induced and because none of the components contains a triangle, it is necessarily the case that all but 2 edges of every region are covered: m − (m − r) = r and every edge is on two regions. Therefore, there is a 2-factor in the dual. Problems. 1. Disprove Hajos’ conjecture. 2. Show a uniquely colorable graph G has a Kk -minor where k = χ(G). 3. Find an algorithm for building a hamiltonian cycle on an imbedding whenever such a cycle exists. 4. Prove Brooks’ Theorem; that is, show χ(G) ≤ ∆(G) or G = Kn or G = C2n+1 .

37

Chapter 8

Answers to some selected exercises 1.1.2. Consider, first, any odd order graph G that is 2r-regular. Then take r components all equal to G. Take a central vertex and swap an edge in each copy of G for two edges incident the central vertex v in a triangle swap. Now remove one edge incident the central vertex. One of the remaining components does not have a perfect matching. P 1.1.3. There exists a tree with degrees d1 , d2 , ..., dn if and only if di = 2n − 2.

m−1 n 1.2.1. For k odd there are no k-cycles. For k even, there are m k−1 k (k − 1)!(k!). 1.2.2. For k odd there are no k-circuits. For k even, there are mk nk closed k-circuits. 1.2.3. In each connected component, there are an even number of odd valence vertices. Recursively pair up the odd valence vertices and construct a path between the vertices. The remaining graph is eulerian in each component and thus has a cycle decomposition. 1.2.4. If 2k + 1 is prime, then this has an especially interesting solution. Use the prime order group Cp and the odd cycles formed by each generator and its inverse. ∆(G)

1.3.2. Consider the graph Pn

.

1.3.3. For magnanimous graphs per(G) = cen(G) = G. 1.3.4. Products of graphs that have this property also share this property. 1.4.2. One sufficient condition is that all the distance classes have cardinality 1. Another sufficient condition is that the distance classes all have cardinality k and that G is k-connected. 38

1.4.3. Suppose the graph G is not hamiltonian and has girth g, then n(G) < (g − 1)(rad(G)). Therefore, if rad(G) ≥ n/(g − 1), then the graph is hamiltonian. P 1.4.4. Show i (i − 2)fi is constant = n − 2 where fi is the number of faces of length i on either side of the cycle. The theorem holds by induction. 1.5.2. List the vertices of the graph in a column to the left and a column to the right. There is a matching by Hall’s Theorem. This matching is equivalent to a 2-factor in the original graph. Strip 2-factors until we are left with a 2-factorization. 1.5.3. This statement does not hold. Consider C4 × Pn . 2.1.1. The graph is perfect so the bandwidth of a line graph is equal to its circumference c − 1. 2.1.2. To generate a super-set of the trees, add a leaf to each vertex of the set for the previous order to generate the trees of order n. (The caterpillars can be generated fairly quickly. Also, consider the problem up to homeomorphism and see Exercise 5.2.1.) 2.1.3. The bandwidth of Pn × Pk is {n, k}. 2.1.4. For all connected graphs G3 is hamiltonian. However, for all n ≥ 6 take the spider SPn with three legs of length at least 2. The graph SPn2 is non-hamiltonian.
Pk=n 2.1.5. Using the method of stars and bars we get k=1 nk /(n − k)! 2.3.1. Consider the two regular orientations of K5 . Count the number of ways of picking two pairs of 2K2 from the orientations up to isomorphism. This count is the number of (2, 2)-regular orientations of 4-regular graphs of order 7: 8. The counting involves case by case analysis using a binary tree of depth 3. Prove the cases are all distinct and exhaustive. Complete the proof by extending the regular orientations to K7 in the natural way. 2.3.4. This should be immediately clear. 2.3.5. Consider (3, 0)2 , (0, 3)2 . 3.1.1. Use the Turan graph of G. This graph necessarily excludes H. 3.1.2. Add edges to form a path, and then add edges in Pk2 , Pk3 , ..., Pkk−1 in order. In the other case, bandwidth n − 1 cannot be achieved until we have a complete graph. Bandwidth n − 2 cannot be obtained until the graph has enough edges so that the complement is contained in a P3 and so on; look at the complement of the powers of the path and then construct threshold graphs that have complements contained in their complements.

39

3.1.3. Consider (s − 1)(t+1) , (t + 1)(s−1) . 3.1.4. The answer is n(G) = 6. 3.1.5. The lower bound is ⌈ m(G)−m(G∩H) ⌉ and is achieved by a long cycle and a 2 triangle factor for 3|n. The upper bound is c1 + c2 − 2 where c1 and c2 are the respective component numbers of the two 2-factors. This is achieved when the two factors have relatively prime length cycles of equal length. 1 ∩T2 ) 3.1.6. The answer is ⌈ m(T1 )−m(T ⌉. 2

3.1.7. Give the lower bound above. In all cases, this suffices. Consider (G ∪ H) − (G ∩ H). 5.2.1. Start with the label n if k is incident k − t, insert k to the immediately to the left of k − t. There are (n − 1)! rooted trees of each order. 5.2.2. Consider a pair of triangles. 5.2.3. The irregularity strength is given by λ(G) in both cases. In the case of a path, label 1,1,2,2,3,3,...,λ. In the case of a cycle use the same labeling, but skip a label so λ + 1 is unique. 5.2.5-6. The answers are similar. Suppose the diameter of a graph G is ≥ 3, then take H = [N [v]] ∪ [N [w]] ∪ kK2 ⊂ G of maximum size. We get that the edges incident the matching are saturated by other edges in the matching of the two stars, or otherwise the matching is not of maximum size. Therefore, G has size 12 k(n − 1) + t2(n − 1 − t1 − k) + t1 (n − 1 − t2 − k); we assume H has maximum size in G. Simplifying yields 1 1 1 (k + 2t1 + 2t2 )n − (k + 2t1 + 2t2 ) − k(1 + 2t1 + 2t2 ) − 2t1 t2 . 2 2 2 Now take all the possible values of k and t1 , t2 that would optimize m(G), under the condition k + t1 + t2 = n − 1. If k = 21 (n − 1), then t1 = t2 = 1 4 (n − 1) and we get m(G) ≤

3 3 1 1 1 n(n − 1) − (n − 1) − n(n − 1) − (n − 1)2 < n(n − 1). 4 4 4 8 2

If k = n − 1 we get m(G) ≤ m(G) ≤ 21 (n − 1)2 .

1 2 (n

− 1)2 . If t1 = t2 =

1 2 (n

− 1) we get


5.2.7. Suppose G has girth 4 and size n2 . Then G has an ascending subgraph decomposition. The proof is by induction on n. Form a double-star matching H and invoke induction. Add edges to the ascending subgraph decomposition using singleton edges from the graph H and then repeating until the edges of the graph H are exhausted. If it ever takes place that we add an edge from H to Gi such that G′i 6⊂ G′i+1 , we have added an edge to a star in Gi and a disjoint matching to Gi+1 . Then consider Gi ∪ Gi+1 40

and strip an edge from Gi+1 and add it to Gi so that Gi−1 ⊂ G′i+1 ⊂ G′i . (Form a Gi -augmenting path. Either there is a free K2 , or we can use the augmenting path to add a K2 without affecting the degree sequence of either graph except to move the K2 from G′i+1 to G′i , forming G′′i and G′′i+1 respectively. It follows that G′i−1 ⊂ G′i (= G′′i ) ⊂ G′′i+1 . Make the initial K2 -swap and continue correcting the sequence until we reach i + 2 = n − 1 when this occurs.) Then continue removing edges from H. In the case that there is no double-star in G, the graph has been reduced to a star forest. It has been demonstrated that star-forests have star-forest ascending subgraph decompositions. It may be the case that H ′ , the edge-reduction of H has n− 1 < m(H ′ ) ≤ 2n− 3 at some stage of the edge-stripping process. In this case, we remove a sequence of edges, then return to the top of the order and begin again. 5.2.8. Without loss of generality, either H or
G is such that if we remove the minimal number of arcs m, where m 2 ≥ m(H ∪ G) and perform an ascending subgraph decomposition, then if m′ is the number of {Hi } that contain arcs, m > m′ . Replace the m arcs in sequence to get an ascending subgraph decomposition of the graph H ∪ G.

5.2.9. Let G be a graph of size n2 < m(G) ≤ n+1 2 . Then G has a star-matching or 2-star-matching of size n. Use this graph to find a star-forest of size n in G. Then use the existence of this star-forest to find a sparsest such star-forest H. Perform an ascending subgraph decomposition of G − H. Add the K2 at the tail of the decomposition to H and work down the decomposition adding disjoint K2 s to the decomposition. If it ever takes place that we can not add an edge from H to Gi such that G′i ⊂ G′i+1 , we have cannot add a disjoint K2 . So add an edge to a star in Gi and a disjoint matching to Gi+1 . Take Gi ∪ Gi+1 and form a Gi -augmenting path. Either there is a free K2 or we can use the augmenting path to add a K2 without affecting the degree sequence of either graph except to move the K2 from G′i+1 to G′i . This gives G′i−1 ⊂ G′i (= G′′i ) ⊂ G′′i+1 . Make the initial K2 -swap and continue correcting the sequence until we reach i + 2 = n − 1 when this occurs.

41

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