Arithmetic Graphs

IN T RODUCT ION Graph Theory is a major area of combinatorics and during recent decades, Graph Theory has developed into all major areas of mathematics. One of the beauties of Graph Theory is that it depends very little on other branches of mathematics. In addition to it’s growing interest and importance as mathematical subject, it has application to many fields including computer science and chemistry. The paper written by Leonhard Euler who is known as the father of Graph Theory, on the Seven Bridges of Konigsberg is regarded as the first paper in the history of Graph Theory. It was published in 1736. This paper, as well as the one written by Vandermonde on the Knight Problem carried on with the Analysis Situs initiated by Leibniz. Euler’s formula relating the number of edges, vertices, and faces of a convex polyhedron was studied and generalized by Cauchy and L’Huillier, and is at the origin of Topology. The term Graph was introduced by Sylvester in a paper published in Nature in 1878. Graph labelling is an important branch of Graph Theory. A graph labelling is an assignment of integers to the vertices , edges or both, subject to certain conditions. Graph labellings were first introduced in the late 1960s. Most graph labelling methods traced their origin to the one introduced by Rosa in 1967, or the one given by Graham and Sloane in 1980. Labelled graphs serves as useful mathematical models for a broad range of applications such as Coding theory, including the design of good radar type codes, synch-set codes, missile guidance codes and convolutional codes with optimal autocorrolation properties. They facilitate the optimal nonstandard encoding of integers. Labelled graphs have also been applied in determining ambiguities in X-ray crystallographic analysis, to design a communication network addressing system, in determining optimal circuit layouts and radio astronomy problems etc. Labellings are of various type and one of them namely additive numbering is an important concept in graph labelling. Additive numbering is further divided and Arithmetic Labeling is a special type of additive numbering. In this dissertation we concentrate on Arithmetic Labelings. The notion of (k,d)-arithmetic numbering of graph was introduced by B.D Acharya and S.M Hegde.

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An Overview Of The Dissertation This dissertation Arithmetic Graphs is a brief discussion on Arithmetic Labelling and properties regarding it. The entire section has been divided into six chapters. The first chapter consists of some basic definitions and results in Graph Theory. The concept of Additive Numbering and Arithmetic Graphs are included in the second chapter. It also consists of some basic definitons such as Indexable, Strongly Indexable ...etc and some priliminary results in Arithmetic Labelling. In the third chapter we have made a short study on arithmetic and other related numberings such as additively (k,d)-sequential numbering, geodetic numbering, graceful numbering and (k,d)-balanced numbering of graph. For all odd number of vertices, the class of cycle related graphs has a (k,d)-arithmetic numbering for some particular values of k and d. A study of this property is done in the fourth chapter. In the fifth chapter we have shown that a class of trees called Tp -trees (transformed trees) and the subdivision S(T) of a Tp -tree have an arithmetic labeling. In the last chapter, our discussion is on some special classes of graphs that are amenable to arithmetic numbering. ? ?

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PRELIMIN ARIES

This chapter consists of some basic definitions and results that are used for the subsequent development of the theory. Definition 1.1 A graph G=(V(G),E(G)) consists of two finite sets V(G),the vertex set of the graph, which is a nonempty set of elements called vertices and E(G), the edge set of the graph, which is possibly empty set of elements called edges such that each edge ‘e’ in E(G) is assigned an unorderd pair of vertices (u,v) called end vertices of ‘e’. The number of vertices and the number of edges in a graph are respectively called the order and size of the graph. If the end vertices of an edge are the same, it is called a loop. If two or more edges have the same end vertices, then the edges are said to be parallel. Definition 1.2 Two vertices which are joined by an edge are said to be adjacent or neighbours. The set of all neighbours of a fixed vertex v of G is called the neighbourhood set v and is denoted by N(v). Definition 1.3 A graph G is called simple if it has no loops or parallel edges. A complete graph is a simple graph in which each pair of distict vertices is joined by an edge. A complete graph with n vertices is denoted by Kn . Definition 1.4 The degree d(v) of a vertex v of a graph G is the number of edges of G incident with v, counting each loop twice, i.e, it is the number of times v is an end vertices of an edge. If for some positive integer k, d(v)=k for every vertex v of a graph G then G is called k-regular.

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Definition 1.5 Let G be a graph, if the vertex set V(G) can be partitioned into two nonempty subsets X and Y, (i.e, X ∪ Y =V(G) and X ∩ Y = φ), in such a way that each edge of G has one end in X and one end in Y then G is called bipartite. The partition V=X ∪ Y is called a bipartition of G. THEOREM 1.6 For any graph G with e edges and n vertices v1 ,v2 ,...vn

n X

d(vi ) = 2e. 

i=1

Definition 1.7 A walk W in a graph G is a finite sequence W = v0 e1 v1 e2 v2 ...ek vk whose terms are alternatively vertices and edges such that each edge ei , 1 ≤ i ≤ k, has ends vi−1 , vi . W is called a v0 − vk walk.v0 is called the initial vertex and vk is the terminus of the walk W. v0 v1 v2 ...vk−1 are called the internal vertices. The number of edges in the walk is called the length of the walk. A u-v walk is said to be closed when u=v and open otherwise. If the edges of a walk W are distinct then W is called a trail. If the vertices are distinct, it is called a path. A path with n vertices is denoted by Pn . Definition 1.8 Two vertices u and v of a graph G are said to be connected if there is a path joining them. A graph G is called a connected graph if any two vertices of it are connected. Definition 1.9 Let G be a simple graph with n vertices the complement G of G is defined to be the simple graph with same vertex set as G and where two vertices u and v are adjacent precisely when they are not adjacent in G. Definition 1.10 A non trivial closed trial in a graph G is called a cycle if its origin and internal vertices are disjoint. A cycle with n edges is called a n-cycle and is denoted by Cn . Definition 1.11 An Euler tour of a graph G is a closed walk of G which includes every edge of G exactly once. A graph G is called Eulerian if it has an Euler tour. THEOREM 1.12 A connected graph G is Eulerian if and only if the degree of every vertex is even.  4

Definition 1.13 A graph G is called acyclic if it contains no cycles. A connected acyclic graph is called a tree. Definition 1.14 A graph G is planar if there exists a drawing of G in the plane in which no two edges intersect in a point other than a vertex of G, where each edge is a simple arc. Such a drawing of a planar graph G is called a plane representation of G. In this case we also say that G has been embedded in the plane. A plane graph is a planar graph that has alreay been embedded in the plane. Definition 1.15 Subdivision graph S(G) of a graph G is obtained by subdividing every edge of G exactly once. ? ?

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ADDIT IVE N UMBERIN GS 2.1 Vertex Functions and Induced Edge Functions Given a graph G=(V,E), the set N of nonnegative integers, a subset A of N and a commutative binary operation ∗ : N × N → N , every vertex function f : V (G) → A induces an edge function f ∗ : E(G) → N such that f ∗ (uv) = f (u) ∗ f (v) for all uv ∈ E(G) Here we concentrate on vertex function f : V (G) → A; A ⊆ N which induces an edge function f ∗ : E(G) → N defined by f ∗ (uv) = f (u) + f (v) for all uv ∈ E(G). Such vertex functions are said to be additive and so we denote f ∗ of f as f + . Definition 2.1.1 An additive numbering of a graph G=(V,E) is an injective additive vertex function f such that the induced edge function f + is also injective. Here we use A(G) to denote the set of all additive numberings of G. Example 2.1.2

Fig 2.1 Consider G with V (G) = {v1 , v2 , v3 , v4 } and E(G) = {v1 v2 , v1 v3 , v2 v4 , v3 v4 } Define 0 0 f : V (G) → A where A = {0, 1, 4, 3} and f (v1 ) = 1, f (v2 ) = 4, f (v4 ) = 3, f (v3 ) = 0. Then f + E(G) :→ N become f + (v1 v2 ) = 5, f + (v2 v4 ) = 7, f + (v3 v4 ) = 3, f + (v1 v3 ) = 1. Clearly vertex function f and the induced edge function f + are injective. Therefore f ∈ A(G). We adopt the following notations

f (G) = {f (u) \ u ∈ V (G)} f ∗ (G) = {f ∗ (x) \ x ∈ E(G)} fmin (G) =

min f (u) u∈V (G)

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+ fmin (G) =

fmax (G) = + fmax (G) =

Θ(G) =

min f + (x) x∈E(G)

max f (u) u∈V (G)

max f + (x) x∈E(G)

min fmax (G)

f ∈A(G)

An additive numbering f of G is said to be minimal if fmax (G) = Θ(G) and let AΘ (G) denote the set of all minimal additive numberings of G. Clearly for any graph G, the set A(G) is infinite and AΘ (G) is finite. THEOREM 2.1.3 For any graph G and any f ∈ AΘ (G), there exist a vertex u of G such that f(u)=0. Proof For any graph G and for any f ∈ AΘ (G), fmax (G) = Θ(G). Moreover, the ranges of f and f + are the set of distinct nonnegative integers. If possible assume that there exists no vertex v in G such that f(v)=0. Let u ∈ V (G) such that f (u) = fmin (G). Substract r ∈ N from each {f (v)\v∈ V (G)} such that f(u)-r=0. Then we get another additive numbering g such that g(v)=f(v)-r ∀v ∈ V (G) Then clearly gmax (G) < Θ(G), which is a contradiction. Therefore there exists a vertex u ∈ V (G) such that f(u)=0.  COROLLARY 2.1.4 For any (p,q) graph G, Θ(G) ≥ p − 1 and this bound is best possible. Proof Since G has p vertices, we have by above theorem, for any f ∈ AΘ (G) their exist a u ∈ V (G) such that f(u)=0. Each f ∈ AΘ (G) is injective, since |V(G)|=p, then the range of f has p distinct nonnegative integers including 0. Therefore fmax (G) ≥ p − 1, ∀f ∈ AΘ (G). But f ∈ AΘ (G) ⇒ fmax (G) = Θ(G). Therefore Θ(G) ≥ p − 1. To show that this bound is best possible, consider the join G = K2 + K p−2 for any integer p ≥ 3. Let the vertices of K2 be labeled by u1 and up and those of K p−2 be labeled by u2 , u3 ...up−1 .

Fig 2.2 7

Clearly G has 2p-3 edges. Define a map by letting f (ui ) = i − 1 , 1 ≤ i ≤ p. Clearly f is injective. Then f + (u1 up ) = p − 1 f + (u1 ui ) = i − 1;

2≤i≤p−1

f + (up ui ) = p − 2 + i;

2≤i≤p−1

So that f + (G) = {1, 2, 3, ..2p − 3}. That is the range of f + is distinct nonnegative integers, where 2p-3=|E(G)| Clearly f is an additive numbering of G and which is minimal Therefore fmax (G) = Θ(G) = p − 1.  If f is an additive vertex function we know that for any x = vi vj ∈ E(G), f + (x) = f (vi ) + f (vj ), vi , vj ∈ V (G) Therefore X X f + (x) = d(u)f (u) x∈E(G)

u∈V (G)

Thus we got a theorem, which is a generalisation of “First Theorem of Graph Theory” THEOREM 2.1.5 For any graph G=(V,E) and for any additive vertex function f : V (G) → N X X f + (x) = d(u)f (u) .  x∈E(G)

u∈V (G)

We have for any(p,q) graph G, the sum of vertex degrees equals 2q. If G is a γ -regular (p,q) graph then for any additive vertex function f of G=(V,E) X X f + (x) = γ f (u) x∈E(G)

u∈V (G)

In particular, the sum of the values of edges of any cycle in G is twice the sum of the values of the vertices of the cycle COROLLARY 2.1.6 If G is an Eulerian graph and f is any additive vertex function of G then the sum of the X values of edges of G is even, That is, f + (x) ≡ 0(mod2) x∈E(G)

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Proof We have,

X

f + (x) =

x∈E(G)

X

d(u)f (u). But G is Eulerian. Every vertex of G is of

u∈V (G)

even degree. That is, d(u) is even ∀u ∈ V (G). X Therefore, d(u)f (u) is even. =⇒

X

u∈V (G) +

f (x) is even.

x∈E(G)

=⇒

X

f + (x) ≡ 0(mod2). 

x∈E(G)

2.2 Indexable Graphs Definition 2.2.1 A graph G for which Θ(G)=|V(G)|-1 is said to be indexable and any minimal numbering of such graph will be called indexer. The following graphs are examples of indexable and nonindexable graphs. Example 2.2.2: Indexable Graphs

Fig 2.3

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Example 2.2.3: Non indexable Graphs

Fig 2.4 Observe that the indexers of G = K2 + K p−1 given in the proof of corollary 2.1.4 and G = K1,p−1 in Figure 2.3(a) have an additional property that f + (G) = {1, 2, 3...q} where q=|E(G)|. Definition 2.2.4 An additive numbering f of a (p,q) graph G is said to be a strong indexer if f (G) = {0, 1, 2...p − 1} and f + (G) = {1, 2, ...q} and if G admits such a numbering. G is called strongly indexable graph. Example 2.2.5:Strongly Indexable Graph

Fig 2.5 Definition 2.2.6 A (p,q) graph G=(V,E) is said to be strongly k-indexable if there exists a bijective labeling f : V (G) −→ {0, 1, 2, ..., (p − 1)} such that induced edge labeling f + : E(G) −→ {k, k + 1, k + 2, ..., k + (q − 1)} is also bijective. The above graph G (Fig 2.5) is an example of strongly 1-indexable graph. ? ?

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ARIT HMET IC AN D OT HER RELAT ED N UMBERIN GS 3.1 Arithmetic Numbering Given a (p,q) graph G and f ∈ A(G), we say that f is (k,d)-arithmetic if f + (G) = {k, k + d, k + 2d...k + (q − 1)d}, where k and d are given positive integers. Let Ak,d (G) denote the set of all (k,d)-arithmetic numberings of G. Further, G is said to be (k,d)-arithmetic, if Ak,d (G) 6= φ. Given k, d, (d ≥ 1), if Ak,d (G) 6= φ, then Ak,d (G) is finite because there are only a finite number of partitions of k into two distinct parts. The two parameters of any (k,d)-arithmetic graph G are

Θk,d (G) = 0

Θk,d (G) =

min

f ∈Ak,d (G)

fmax (G)

max fmax (G)

f ∈Ak,d (G)

An additive numbering f ∈ Ak,d (G) is said to be minimal or maximal according to 0 whether fmax (G) = Θk,d (G) or fmax (G) = Θk,d (G). Clearly, for any (k,d)-arithmetic graph G 0

Θ(G) ≤ Θk,d (G) ≤ Θk,d (G) ≤ k + (q − 1)d. where the last equality is attained by the star K1,p−1 assigning 0 to its center and k, k+d, ...k+(p-2)d to its end vertices at random in one to one manner. The following Figure 3.1 is an example of K1,p−1 with k=1 and d=1.

Fig 3.1

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Definition 3.1.1 A graph G is said to be arithmetic, if it is (k, d)-arithmetic for some positive integers k and d. Example 3.1.2

Fig 3.2 The values of edges of G1 can be arranged in arithmetic progression 1, 2, 3, 4, 5, 6. Thus G1 is (1, 1)-arithmetic. Example 3.1.3

Fig 3.3 The values of edges of G2 can be arranged in arithmetic progression 2, 4, ..., 24. Thus G2 is (2,2)-arithmetic. The following result gives a necessary condition for a graph to be (k, d)-arithmetic when k and d take certain values. THEOREM 3.1.4 If a (p, q) graph G=(V, E) has a (k, d)-arithmetic numbering f with k and d are not simultaneously even,then there exists a partition of V(G) into two nonempty subsets π and

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η such that the number of edges joining vertices of π with those of η is exactly 

 q+1 , if k and d are both odd 2 hq i 2

, if k is even and d is odd

(1)

(2)

and q, if k is odd and d is even.

(3)

Proof We partition the vertex set V(G) into two nonempty subsets. Let π = {u ∈ V (G) | f (u) is even} and η = V (G) − π be a partition of the vertex set. Since G has a (k,d)-arithmetic numbering f the edges of G can be listed as f + (G) = {k, k + d, k + 2d, ...k + (q − 1)d}. Since k and d are not simultaneously even, the number of odd numbers in f + (G) gives the number of edges joining the vertices of π and η. Clearly π and η are nonempty subsets of V(G). Since any edge e of G joins a vertex of π with one of η ⇔ f + (e) ≡ 1(mod2), case(a): k and d are both odd Here {k,k+2d, k+4d..., k+(q-2)d, or k+(q-1)d} odd numbers, accoding to q is even or    q+1  odd. Then clearly, there are exactly q−1 + 1 = edges joining vertices of π with 2 2 those of η. case(b): k is even and d is odd In this case {k+d,k+3d,k+5d,....k+(q-2)d, or k+(q-1)d } are odd numbers, as q is odd   or even. Then clearly there are 2q edges joining vertices of π with those of η. case(c): k is odd and d is even In this case {k,k+d,k+2d,.....k+(q-1)d} are odd. i.e. f + (G) contains only odd terms. Therefore, the number of edges joining π with the number of edges joining η is exactly q.  From the above theorem, we can easily deduce that if k is odd and d is even G is bipartite. Since π and η are nonempty , the number of edges joining vertices of π and those of η is exactly q. That is, for any edge one end vertex is from π and other is from η . Now we discuss certain properties of (k,k)-arithmetic (p,q) graphs. THEOREM 3.1.5 Let G be a connected (k,k)-arithmetic (p,q) graph. Then for any f ∈ Ak,k (G),0 ∈ f (G) if and only if k | f (u) ∀u ∈ V (G). 13

Proof Let f ∈ Ak,k (G). Then f + (G) = {jk | 1 ≤ j ≤ q}, that is, f + (G) = {k, 2k, 3k, ..qk}. Suppose k|f(u) ∀u ∈ V (G). Choose the edge xy ∈ E(G) such that f + (xy) = k ⇒ either f(x)=0 or f(y)=0 . Therefore 0 ∈ f (G). Conversly suppose that 0 ∈ f (G) ⇒ there exist u0 ∈ V (G) such that f (u0 ) = 0. Then ∀w ∈ N (u0 )={v ∈ V (G)|u0 v ∈ E(G)} . We have, f (u0 ) + f (w) = f + (u0 w)=a.k for some positive integer ’a’. This yields f (w) = aw k, since f (u0 ) = 0. Clearly k | f (w) 0 ∀w ∈ N (u0 ) ∪ {u0 }. Now fix any w0 ∈ N (u0 ). Then for any w ∈ N (w0 ) − ({u0 }) ∪ N (u0 ), 0 0 we have f (w0 ) + f (w ) = f + (w0 w ) = b.k for some positive integer b ≥ 1 so that 0

f (w ) = bk − f (w0 ) = bk − aw0 k = (b − aw0 )k

0

0

, where (b − aw0 )k > 0 since f is arithmetic numbering. Clearly  k | f (w ).Since w was [ arbitrary, by choice we have k | f (w) ∀w ∈ {u0 } ∪ N (u0 ) ∪  N (y). Continuing y∈N (w0 )

this way we can show that the image of all the vertices of G under f are multiples of k.  From this theorem we can easily deduce that for a (1,1)-arithmetic graph G 0 ∈ f (G) ∀f ∈ A1,1 (G). Also if G is a (2,2)-arithmetic graph, then for any f ∈ A2,2 (G), f(G) consists of only even integers including 0. THEOREM 3.1.6 Let G be a connected bipartite (p,q) graph that is not star. If G is (k,k)-arithmetic, then for any f ∈ Ak,k (G), 0 ∈ / f (G) Proof Let f ∈ Ak,k (G) and let A = {u1 , u2 , ...ua } and B = {v1 , v2 , ...vb , } costitute the parts of a bipartition of G. Suppose that 0 ∈ f (G). Without loss of generality we can assume that f (u1 ) = 0. Since f + (G) = {k, 2k, 3k...qk}, we must have f (vj ) = k for some vj ∈ N (u1 ). Without loss of generality let j=1. Let uv ∈ E(G) such that 2k = f + (uv) = f (u) + f (v) But we have by Theorem 3.1.5 if G is a connected (k, k)-arithmetic (p, q) graph and f ∈ Ak,k (G) then, 0 ∈ f (G) ⇐⇒ k | f (u) ∀u ∈ V (G). Applay this theorem we get k divides both f(u) and f(v). Then there arise two possibilities. Either both f(u)=k and f(v)=k or either f(u) or f(v) is 0 and the other is 2k. The first case is not possible, since f is injective and 14

we already chose f (v1 ) = k. So latter case holds. Thus we have f(u)=0 and f(v)=2k. Therefore u = u1 and we can choose v = v2 . i.e. f (v2 ) = 2k. Let xy ∈ E(G) be such that 3k = f + (xy) = f (x) + f (y). Again by Theorem 3.1.5, k divides both f(x) and f(y). There arise two possibilities. Either one of f(x) and f(y) is k and other is 2k or one of f(x) and f(y) is 3k. Clearly the first possibility cannot arise as f is injective and vertices labelled by k and 2k have alredy occurred. So the latter case must hold. Then without loss of generality we can assume that f(x)=0 and f(y)=3k. But then injectivity of f forces x = u1 so that y = v3 . Continuing like this we can see thatN (u1 ) = {v1 , v2 , ...vt } where t is the degree of u1 in G. Then f (vj ) = jk for each j ,1 ≤ j ≤ t. Since G is not a star and is connected, it follows that | A |≥ 2 so that (t + 1)k ∈ f + (G). Let ui vj ∈ E(G) be such that (t + 1)k = f + (ui vj ) = f (ui ) + f (vj ) This yields f (ui ) = (t + 1)k − f (vj ) ≤ tk as f (vj ) ≥ k. But this is a contradiction to the fact that f (N (u1 )) = {jk | 1 ≤ j ≤ t} and that f is injective. Hence the proof.  From the above theorem we can easily conclude that no connected bipartite graph, except the star, is (k, k)-arithmetic for k=1 or k=2. Also we have any (1,1)-arithmetic or (2,2)-arithmetic graph is either a star or a triangle THEOREM 3.1.7 Let G =(V,E) be an Euilerian (p,q) graph and suppose that G is (k, d)-arithmetic. Then q(2k + (q − 1)d) ≡ 0(mod 4) Proof G=(V, E) is Eulerian graph. So, the degree of every vertex of G is positive even integer. X X X For any f ∈ Ak,d (G) f + (x) = d(u)f (u). Therefore f + (x) is a positive x∈E(G)

u∈V (G)

X x∈E(G) even integer say 2r, where r is a positive integer. Therefore f + (x) = 2r. x∈E(G) +

But f (G) = {k, k + d, k + 2d, ...k + (q − 1)d} X q Then LHS is given by f + (x) = (2k + (q − 1)d) = 2r 2 x∈E(G)

=⇒ q(2k + (q − 1)d) = 4r =⇒ q(2k + (q − 1)d) ≡ 0(mod 4).  If G is an Eulerian (p, q) graph with q ≡ 2(mod 4) is (k,d)-arithmetic, then d ≡ 0(mod 2), since q(2k + (q − 1)d) ≡ 0(mod 4) q(2k + (q − 1)d) = 4r; r is a positive integer.

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COROLLARY 3.1.8 An Euilerian (p, q) graph with q ≡ 3(mod 4) is (k, d)-arithmetic when k is even and d is odd. Proof Given q ≡ 3(mod 4) =⇒ q − 3 is a multiple of 4. =⇒ q is odd and (q − 1) ≡ 2(mod 4) If G is (k, d)-arithmetic, then by above theorem q(2k+(q-1)d is a multiple of 4, also given that k is even and d is odd. =⇒ q(q − 1)d is a multiple of 4. =⇒ qd is a multiple of 2. This is a contradiction, since q and d are odd.  3.2 Additively (k, d)-Sequential Numbering Definition 3.2.1 Given a (p, q) graph G=(V, E) we call f ∈ A(G) an additively (k, d)-sequential numbering of G if, f (G) ∪ f + (G) = {k, k + d, k + 2d, ...k + (p + q − 1)d} and G is said to be additively (k, d)-sequential if it admits an additively (k, d)-sequential numbering Example 3.2.2 (1)

Fig 3.4 We define f : V (G) −→ N as shown in Figure 3.4. Then f (G1 )∪f + (G1 ) = {2, 4, 6, 8....26, 28}. So G1 is additively (2, 2)- sequential. (2)

Fig 3.5 16

We define f : V (P5 ) −→ N as in Figure 3.5. Then f (P5 ) ∪ f + (P5 ) = {1, 2, 3...9}. So P5 is additively (1, 1)-sequential. Since k ≥ 1, from the definition, no additively (k, d)-sequential numbering assigns ’0’ to any vertex of G. We denote Sk,d (G) as the set of additively (k, d)-sequential numbering of G. When d=1, these definition reduces to a special class k- sequentially additive(numbering of graph). In fact, there is a one-to-one correspondence between the set of additively (k, d)sequential graph and the set of k- sequentially additive graph (numberings of graph). THEOREM 3.2.3 Let G be an additively (k, d)-sequential graph and F ∈ Sk,d (G). If d | k then the function f defined by f (u) = F (u) for u ∈ V (G) is an additively (r, 1)-sequential(or an “r- sequentially d additive”) numbering of G, where r is some integer such that k=rd. Conversly given any r- sequentially additive numbering f of G, the function F defined by F (u) = d · f (u), ∀u ∈ V (G), is an additively (rd, d)-sequential numbering of G. Proof Given F ∈ Sk,d (G), then F (G) ∪ F + (G) = {k, k + d, ...k + (p + q − 1)d}. But given d | k =⇒ k = rd, where r is some integer. Then F (G) ∪ F + (G) = {rd, rd + d, ...rd + (p + q − 1)d} = {rd, (r + 1)d, ...[r + (p + q − 1)d]}

Then the function f defined by f (u) = For any xy ∈ E(G) we have

F (u) d

∀u ∈ V (G)

f + (xy) = f (x) + f (y) F (x) F (y) = + d d + F (xy) = d Then F (G) F + (G) ∪ d d = {r, r + 1, r + 2, ...r + (p + q − 1)}

f (G) ∪ f + (G) =

This shows that f is an additively (r, 1)-sequential numbering of G. 17

Conversly suppose that f is a r-sequentially additive numbering of G. Then f (G) ∪ f + (G) = {r, r +1, r +2, ...r +(p+q −1)}. Consider the function F defined by F (u) = d·f (u) for u ∈ V (G). Consider any xy ∈ E(G) we have F + (xy) = F (x) + F (y) = df (x) + df (y) = d(f (x) + f (y)) = df + (xy)

Therefore F (G) ∪ F + (G) = (df (G)) ∪ (df + (G)) = {rd, rd + d, rd + 2d, ...rd + (p + q − 1)d}

This shows that F is an additively (rd, d)-sequential numbering of G.  THEOREM 3.2.4 Let G be an additively (k, d)-sequential graph and f ∈ Sk,d (G). Then the join H = G + K1 is (k, d)-arithmetic. Proof We know that for any f ∈ Sk,d (G) is injective, we must have f (G) ∩ f + (G) = φ. Let V (K1 ) = {v}. Consider the mapping F : V (H) −→ N defined by ( f (u) if u ∈ V (G) F (u) = 0 if u = v Clearly F is a (k,d)-arithmetic numbering of H.  From the above theorem we can see that Pn + K1 are (1,1)-arithmetic. Since Pn is additively (1,1)-sequential. THEOREM 3.2.5 If an odd degree (p, q) graph G is additively (k, d)-sequential then (p + q)(2k + (p + q − 1))d ≡ 0(mod 4) Proof Given that G is an odd degree graph. Consider the join H = G + K1 , which is an even degree graph. Further, since G is additively (k, d)-sequential, by Theorem 3.2.4, H must be 18

(k, d)-arithmetic. Thus, H must be an Eulerian (k, d)-arithmetic graph with q(H)=p+q. But we have, for a (k, d)-arithmetic graph G, q(2k + (q − 1)d) ≡ 0(mod 4). Applying this theorem by replacing q by q(H) we get (p+q)(2k+(p+q−1))d ≡ 0(mod 4).  Our next theorem gives a method to recursively enlarge a given (k, d)-arithmetic numbered graph G to a (k, d)-arithmetic numbered graph H of an arbitrarily high order. THEOREM 3.2.6 Let G be a (p, q) graph having an additively (k, d)-arithmetic numbering f such that the elements of f(G) can be arranged as a subsequence P of an arithmetic progression Q whose last term is that of P (i.e. fmax (G)). Let X = {x1 , x2 , ...xt } = V (Kt ) and let Y = {a1 , a2 , ...ar } be a set with V (G) ∩ Y = φ, where r is a number of terms in Q that are not in P. Then the graph (G ∪ Y ) + K t is a (k, d)-arithmetic graph for any integer t ≥ 1 Proof Let γ : Y −→ (Q − P ) be any bijection. This is possible since | Y |=| Q − P |= r. We have to find a (k, d)-arithmetic numbering on (G ∪ Y ) + K t with t ≥ 1 Define F : V (G) ∪ Y ∪ X −→ N by saying   if u ∈ V (G)  f (u) F (u) = γ(ai ) if u = ai ∈ Y   k + qd − fmin (G) + d(p + r)(i − 1) if u = xi ∈ X We can easily verify that F is a (k, d)-arithmetic numbering of (G ∪ Y ) + K t .  Example 3.2.7 Consider (3, 3)-arithmetic numbered graph G as shown in Figure 3.6(a).

Fig 3.6(a) Then f (G) = {0, 3, 6, 9, 15}. Consider P=(0, 3, 6, 9, 15) and the arithmetic progression Q=(0, 3, 6, 9, 12, 15). Put Y = {a1 } and let X = {x1 , x2 }. Arithmetic numbering of the augmented graph H = G ∪ Y is shown in Figure 3.6(b).

19

Fig 3.6(b) Consider γ : Y −→ (Q − P ) be a bijection. Clearly γ(a1 ) = 12 Now define a map F : V (G) ∪ Y ∪ X −→ N by   if u ∈ V (G)  f (u) F (u) = γ(ai ) if u = ai ∈ Y   k + qd − fmin (G) + d(p + r)(i − 1) if u = xi ∈ X Here r is the number of terms in Q are not in P. Consider the graph H + K 2 , where V (K2 ) = {x1 , x2 } = X. Clearly, F (a1 ) = 12 F (x1 ) = 3 + 6 × 3 − 0 + 3(5 + 1)(1 − 1) = 21 SimilarllyF (x2 ) = 39

Fig 3.6(C) F + (H + K 2 ) = {3, 6, 9, 12, ....54} . Clearly H + K 2 is (3,3)-arithmetic as shown in Figure 3.6(c). 20

Example 3.2.8

Fig 3.7(a) Let G is a (4, 4)-arithmetic graph as shown in Figure 3.7(a), with f (G) = {0, 4, 8, 12, 16, 24} and P = {0, 4, 8, 12, 16, 24}, Q = {0, 4, 8, 12, 16, 20, 24}, Y = {a1 ), X = {x1 , x2 }. Arithmetic numbering of the augmented graph H = G ∪ Y is shown in Figure 3.7(b) Consider γ : Y −→ (Q − P ) be a bijection. Clearly γ(a1 ) = 20.

Fig 3.7(b) Now define a map F : V (G) ∪ Y ∪ X −→ N By   if u ∈ V (G)  f (u) F (u) = γ(ai ) if u = ai ∈ Y   k + qd − fmin (G) + d(p + r)(i − 1) if u = xi ∈ X Here r is the number of terms in Q, which are not in P.

F (a1 ) = 20 F (x1 ) = 4 + 7 × 4 − 0 + 4(6 + 1)(1 − 1) 21

= 32 SimilarllyF (x2 ) = 60

Fig 3.7(c) F + (H + K 2 ) = {4, 8, 12, 16, 20, ......84}. Clearly H + K 2 is (4, 4)-arithmetic as shown in Figure 3.7(c). 3.3 Sequential and Geodetic Numbering Definition 3.3.1 A (k, d, α)-arithmetic graph we mean a (k, d)-arithmetic graph G for which θk,d (G) ≤ α Definition 3.3.2 A (p,q) graph G is said to be sequential, if there exists a positive integer k such that G is (k, 1, α )-arithmetic for α = q or α = q − 1 whether or not G is a tree, respectively. Definition 3.3.3 An injective vertex function f : V (G) −→ N of a (p,q) graph G is said to be k-sequential if the induced edge function gf : E(G) −→ N defined by gf (uv) =| f (u)−f (v) | ∀uv ∈ E(G) such that f (G) ∪ gf (G) = {k, k + 1, k + 2, ...k + p + q − 1}, and a graph is said to be k-sequential if it admits a k-sequential numbering f as defined. Definition 3.3.4 An injective map f : V (G) −→ N is called a geodetic numbering of the graph G, if the induced map gf : E(G) −→ N defined by gf (uv) =| f (u) − f (v) | ∀uv ∈ E(G) is also injective. 22

Let D(G) denote the set of all geodetic numberings of G and Φ(G) = min fmax (G). f ∈D(G)

Then for any connected (p, q) graph G, Φ(G) ≥ q. The maps in D0 (G) = {f ∈ D(G) | fmax (G) = Φ(G)} are said to be minimal. Definition 3.3.5 A graph G is said to be graceful if Φ(G) = q and the minimal geodetic numberings of a graceful graph are called graceful numbering of the graph. For a (p, q) graph G, a graceful numbering is an injective mapping f : V (G) −→ N such that f (G) ⊆ {0, 1, 2...q} and gf (G) = {1, 2, 3...q} and if, G is graceful then it admits such a geodetic numbering on it’s vertices. Definition 3.3.6 Given positive integer k and d a (k, d)-graceful numbering of G is an injective mapping f : V (G) −→ N such that f (G) = {0, 1, 2, ..k + (q − 1)d} and gf (G) = {k, k + d, k + 2d, ...k + (q − 1)d} and G is said to be (k, d)-graceful, if it admits such a numbering. Definition 3.3.7 A (k, d)-graceful graph G is said to be (k, d)-balanced, if, it has a (k, d)-graceful numbering f such that for some positive integer m either f (u) ≤ m and f (v) > m or f (u) > m and f (v) ≤ m for every edge uv. Then we say that f is a (k, d) -balanced numbering of G and m is the characteristic of f denoted by m(f). Example 3.3.8-Geodetic Numbering of a Graph. (a)

Fig 3.8 Consider the injective map f : V (G) −→ N as, f (v1 ) = 0, f (v2 ) = 3, f (v3 ) = 2. gf : E(G) −→ N defined by gf (uv) =| f (u) − f (v) |. Then gf (G) = {1, 2, 3}. Clearly gf is injective. Therefore f : V (G) −→ N is a geodetic numbering of G.

23

(b)

Fig 3.9 f : V (G) −→ N as f (v1 ) = 1, f (v2 ) = 0, f (v3 ) = 2, f (v4 ) = 8, f (v5 ) = 5. Clearly f is injective. Define gf : E(G) −→ N as gf (vi vj ) =| f (vi )−f (vi ) |, then gf (G) = {1, 2, 3, 4, 6} and gf is injective. Therefore f : V (G) −→ N is a geodetic numbering of a graph of G. Example 3.3.9: (k,d)-Graceful Numbering of a Graph (a)

Fig 3.10 Define f : V (G) −→ N as f (v1 ) = 0, f (v2 ) = 1, f (v3 ) = 3, f (v4 ) = 4. Then f is injective. Define gf : E(G) −→ N as gf (vi vj ) =| f (vi ) − f (vi ) |, then gf (G) = {1, 2, 3, 4, } and gf is injective. Clearly G is a (1,1)-graceful. Example 3.3.10: (k, d)-Balanced Numbering of S(K1,3 ) (1) (2, 1) Balanced numbering of S(K1,3 )

Fig 3.11 24

Clearly charecteristic of G1 is 4. (2) (3, 2) Balanced numbering of S(K1,3 )

Fig 3.12 Clearly charecteristic of G2 is 8. THEOREM 3.3.11 Let f be a (k,d) balanced numbering of a bipartite (p, q)graph G. Then the mapping F : V (G) −→ N defined by ( f (u) if f (u) ≤ m(f ) F (u) = k + (q − 1)d + m(f ) + 1 − f (u) if f (u) > m(f ) is an (m(f)+1, d)-arithmetic numbering of G. Proof Let A = {u ∈ V (G) | f (u) ≤ m(f )} and B = V (G) − A. The restriction maps F | A and F | B are clearly injective. Suppose to the contrary that there exist vertices x and y of G such that x ∈ A, y ∈ B and F(x)=F(y). Then by the above definition of F we get k + (q − 1)d + m(f ) + 1 − f (y) ≤ m(f ) =⇒ f (y) ≥ k + (q − 1)d + 1 > k + (q − 1)d This is a contradiction to the definition of a (k, d)-graceful numbering of G. Thus, it follows that F is injective. For any edge uv of G we have u ∈ A and v ∈ B, since A, B is a bipartition of G. Hence F + (uv) = f (u) + k + (q − 1)d + m(f ) + 1 − f (v) = k + (q − 1)d + m(f ) + 1 − (f (v) − f (u)) = k + (q − 1)d + m(f ) + 1 − (k + jd),

0≤j ≤q−1

= (q − 1)d + m(f ) + 1 − jd,

0≤j ≤q−1

= m(f ) + 1 + (q − 1 − j)d,

0≤j ≤q−1

25

It follows, that F + (G) = {m(f )+1, m(f )+1+d, m(f )+1+2d, ..., m(f )+1+(q −1)d}, then F + is also injective. Thus, F is an (m(f)+1,d)-arithmetic numbering of G.  For the (2,1)-balanced numbering f of S(K1,3 ) shown in Figure 3.11, we have m(f)=4 and hence the (5, 1)-arithmetic numbering F using Theorem 3.3.11 for it is displayed in Figure 3.13.

Fig 3.13 Definition 3.3.12 For any positive integer m, let Zm denote the additive group of integers modulo m. Then an additive numbering f of a (p, q)-graph G is called elegant if f (G) ⊆ Zq+1 and f + (G)(mod q + 1) = Zq+1 − {0}, strongly k-elegant numbering if f (G) ⊆ {0, 1, 2, ..., q} and f + (G) = {k, k + 1, ....k + q − 1} and strongly k-harmonious if, f (G) ⊆ {0, 1, 2, ..., q − 1} and f + (G) = {k, k + 1, ....k + q − 1}. Example-3.3.13: Elegant Numbering Consider the graph G on Figure 3.14.

Fig 3.14 26

Clearly it is an elegant numbering. Since q=6, f (G) ⊂ Z7 and f + (G) = {1, 2, 3, 4, 5, 6} = Z7 − {0}. It is also a strongly 1-elegant numbering and strongly 1-harmonious numbering. Hence, the graph G, is said to be elegant (strongly k-elegant, strongly k-harmonious respectively) if, it admits an elegant (strongly k-elegant, strongly k-harmonious respectively) numbering. The following are some straight forward observations. 0.1 For any graph G, the notions of (1, 1)-arithmetic numbering of G and strongly 1-elegant numbering of G are one and same. However, if (k, d) 6= (1, 1) then notion of (k, d)-arithmetic graphs and strongly k-elegant graphs are different. 0.2 Strongly 1-harmonious graphs form a subclass of (1, 1)-arithmetic graphs. ? ?

? ? ? ? ? ? ? ?

27

CYCLE RELAT ED ARIT HMET IC GRAPHS 4.1 Cycle Related Graphs The following are some cycle related graphs. (1) The wheel Wn = Cn + K1 , where Cn cycle of length n, where n ≥ 3. Example 4.1.1

Fig 4.1 (2) The helm Hn is the graph obtained from the wheel Wn by attaching a pendant edge at each vertex of the n-cycle. Example 4.1.2

Fig 4.2 (3) The web graph W(2, n), is the graph obtained by joining the pendant point of a helm Hn to form a cycle and then adding a single pendant edge to each vertex of the outer cycle.

28

Example 4.1.3

Fig 4.3 (4) The generalized web graph W(t, n), is the graph obtained by iterating the process of constructing web graph W(2,n) from the helm Hn , so that the web has t, n-cycles. (5) The crown Cn K1 , is the graph obtained from a cycle Cn by attaching a pendant edge at each vertex of the cycle. Example 4.1.4

Fig 4.4 (6) The generalized web graph without center W0 (t, n), is the graph obtained by removing the central vertex of W(t, n). Example 4.1.5

Fig 4.5 29

(7) We define the graph generalized p-web graph cone as W0 (t, n) + K p where K p is the complement of complete graph with p vertices. Example 4.1.6

Fig 4.6 In this section, we prove that for all odd n, the cycle related Wn , Hnh, W(2,n), h graphs i i (n−1) (3n−1) W(t,n), Cn K1 , W0 (t, n) + K p are (k, d)-arithmetic for k = d and k = d. 2 2 4.2 Arithmetic Labeling of Cyclic Related Graphs THEOREM 4.2.1 For all positive h i integer d and odd n, the generalized web graph W(t, n) is (k, d)-arithmetic (n−1) for k = d. 2 Proof Label the vertices of W(t,n) as follows: Denote the vertices of the innermost cycle of W(t,n) successively as v1,1 ,v1,2 ,v1,3 , ....v1,n . Then denote the vertices adjecent to v1,1 , v1,2 , v1,3 , ....v1,n on the second cycle as v2,1 , v2,2 , v2,3 , ....v2,n respectively and the the vertices adjecent to v2,1 , v2,2 , v2,3 , ....v2,n on the 3rd cycle as v3,1 , v3,2 , v3,3 , ....v3,n and the vertices on the tth cycle as vt,1 , vt,2 , ....vt,n . Next denote the pendant vertices adjacent to vt,1 , vt,2 , ....vt,n as vt+1,1 , vt+1,2 , ....vt+1,n respectively and the centre of the web as v0,0 . Define a labeling f : V (W (t, n)) −→ N such that ( f (u) if u ∈ V (G) F (u) = 0 if u = v 30

f (vm,i ) =

 i−1 ( 2 )d + (m − 1)nd     n−1 i    ( 2 )d + ( 2 )d + (m − 1)nd

( 3n−1 )d + ( i−1 )d + (m − 2)nd 2 2

    (n − 1)d + ( 2i )d + (m − 2)nd    n−1 ( 2 )d + 2tnd

m odd, i odd;

1 ≤ m ≤ t + 1,

1≤i≤n

m odd, i even;

1 ≤ m ≤ t + 1,

2≤i≤n−1

m even, i odd;

2 ≤ m ≤ t + 1,

1≤i≤n

m even, i even;

2 ≤ m ≤ t + 1,

2≤i≤n−1

(4)

m = i = 0.

Because f is injective on every cycle and the maximum vertex value in the mth cycle is less than the minimum vertex value in the (m + 1)th cycle, it is not hard to verify that f defined above is injective. Similarly one can see that f + (W (t, n)) = {k, k + d, ...., k + [(2t + 1)n − 1]d} where k = [ (n−1) ]d. Hence W(t,n) is (k, d)-arithmetic for k = [ (n−1) ]d.  2 2 For example, (4, 1)-arithmetic labeling of W(4, 9) using Theorem 4.2.1 is shown in Figure 4.7

Fig 4.7

31

Remark 4.2.2 For all h positive i integers d and n, the generalized web graph W(t, n), is (k, d)-arithmetic (3n−1) for k = d. (Proof is analogous to Theorem 4.2.1). Therefore the value of k is not 2 uniqe for W(t, n) to be (k, d)-arithmetic. h i For example, using k = (3n−1) d a (13, 1)-arithmetic labeling of W(4, 9) is shown in 2 Figure 4.8.

Fig 4.8 COROLLARY 4.2.3 For n odd, the helm Hn the web graph W(2, n) are (k, d)-arithmetic for k = [ (n−1) ]d. 2 Proof By taking t=1 and t=2 respectively in equation (4), the proof is trivial.  COROLLARY 4.2.4 For n odd, the crown Cn K1 is (k, d)-arithmetic for k = [ (n−1) ]d. 2 Proof Taking t=1 in equation (4) and removing the center of the helm Hn the proof follows.  32

COROLLARY 4.2.5 For n odd, the generalized web graph without centre W0 (t, n) is strongly k-indexable for k = [ (n−1) ].  2 COROLLARY 4.2.6 For n odd, the generalized p-web cone without centre W0 (t, n) + K p is (k,d)-arithmetic for k = [ (n−1) ]d.  2 We have examples for wheels Wn and arithmetic helms Hn when n is even (Following Figure 4.9). But whether the generalized web W(t, n) is (k, d)-arithmetic or not when n even is an open problem.

Fig 4.9 (3,1)-arithmetic labelng of W4 and H4 ? ?

? ? ? ? ? ? ? ?

33

A CLASS OF ARIT HMET IC T REES In this section we prove that a class of trees called Tp -trees (transformed trees) are (k+(q1)d,d)-arithmetic for all positive integers k and d. Also we prove that the subdivision S(T) of a Tp - tree T, obtained by subdividing every edge of T exactly once is (k+(q-1)d, d)arithmetic for all positive integers k and d. (Note that q is the number of edges of T and the subdivision S(T) of a Tp - tree T is not necessarily a Tp - tree) 5.1 Arithmetic Labeling of Transformed Trees (Tp -trees) Definition 5.1.1 Let T be a tree and u0 and v0 be two adjacent vertices in T. Let there be two pendant vertices u and v in T such that the length of u0 − u path is equal to the length of v0 − v path. If the edge u0 v0 is deleted from T and u and v are joined by an edge uv, then such a transformation of T is called an elementary parallel transformation (or an ept) and the edge u0 v0 is called a transformable edge. If by a sequence of ept’s T can be reduced to a path then T is called a Tp -tree (transformed tree) and any such sequence regarded as a composition of mappings (ept’s) denoted by P, is called a parallel transformation of T. The path, the image of T under P, is denoted as P(T). A Tp -tree and a sequence of two ept’s reduces to a path are illustrated in the following Figure 5.1.

Fig 5.1 34

THEOREM 5.1.2 Every Tp -tree T is (k+(q-1)d, d)-arithmetic for all positive integers k and d. Proof Let T be a Tp -tree with n+1 vertices. By the definition of a Tp -tree there exists a parallel transformation P of T for the path P(T), we have (i) V(P(T))=V(T) (ii) E(P(T))=(E(T)-Ed )∪Ep where Ed is the set of edges deleted from T and Ep is the set of edges newly added through the sequence P=(P1 , P2 , ....Pk ) of the ept’s Pi used to arrive at the path P(T). Clearly Ed and Ep have the same number of edges. Then denote the vertices of P(T) successively as v1 , v2 ,... vp starting from one pendent vertex of P(T) right up to other. Define f : V (P (T )) −→N by ( f (vi ) =

(i−1)d 2

for odd i,

k + (q − 1)d +

h

(i−2) 2

i

1≤i≤n+1

d for even i, 2 ≤ i ≤ n + 1

where k and d are positive integers and q is the number of edges of T. Clearly f is a (k+(q-1)d, d)-arithmetic labeling of P(T). Let vi vj be an edge in T for some indices i and j, 1 < i < j ≤ n + 1 and let P1 be an ept obtained by deleting this edge and adding the edge vi+t vj−t where t is the distance of vi from vi+t and the distance of vj from vj−t . Let P be a parallel transformation of T that contains P1 as one of the constituent ept’s. Since vi+t vj−t is an edge in the path P(T) it follows that i+t+1=j-t ⇒ j = i + 2t + 1. The value of the edge vi vj is f + (vi vj ) = f + (vi vi+2t+1 ) = f (vi ) + f (vi+2t+1 )

(5)

If i is odd and 1 ≤ i ≤ n, i h i h then (i+2t+1−2) d + k + (q − 1)d + d f (vi ) + f (vi+2t+1 ) = (i−2) 2 2 = k + (q − 1)d + (i + t − 1)d.

(6)

If i is even and 2 ≤ i ≤ n, then i h i h (i+2t+1−1) d + d f (vi ) + f (vi+2t+1 ) = k + (q − 1)d + (i−2) 2 2 = k + (q − 1)d + (i + t − 1)d

35

(7)

Therefore, from (5), (6), (7), we get f + (vi vj ) = k + (q − 1)d + (i + t − 1)d ∀i, 1 ≤ i ≤ n

(8)

The value of the edge vi+t vj−t is f + (vi+t vj−t ) = f (vi+t ) + f (vj−t ) = f (vi+t ) + f (vi+t+1 )

(9)

If i+t is odd, then h i h i (i+t+1−2) f (vi+t ) + f (vi+t+1 ) = (i+t−1) d + k + (q − 1)d + d 2 2 = k + (q − 1)d + (i + t − 1)d

(10)

If i+t is even, then h i h i (i+t−2) f (vi+t ) + f (vi+t+1 ) = (i+t+1−1) d + k + (q − 1)d + d 2 2 = k + (q − 1)d + (i + t − 1)d

(11)

Therefore, from (9), (10), (11), we get f + (vi+t vj−t ) = k + (q − 1)d + (i + t − 1)d

(12)

From (8) and(12), f + (vi vj ) = f + (vi+t vj−t ). Hence, f is a (k+(q-1)d, d) -arithmetic labeling of T.  For example, a (14, 1) -arithmetic labeling of a Tp -tree, using Theorem 5.1.1 is shown in the following Figure 5.2.

Fig 5.2

36

5.2 Arithmetic Labeling of Subdivision of a Tp -Tree THEOREM 5.2.1 If T is a Tp -tree with q edges then the subdivision tree S(T) is (k +(q −1)d, d)-arithmetic for all positive integers k and d. Proof Let T be a TP -tree with n vertices and p edges. By the definition of a Tp -tree there exists a parallel transformation P of T so that we get P(T). Denote the succession vertices of P(T) as v1 , v2 , ...., vn starting from one pendant vertex of P(T) right up to other and preserve the same for T. Now construct the sundivision tree S(T) of T by introducing exactly one vertex between every edge vi vj of T and denote the vertex as vi,j . Let vmx vhx , x = 1, 2....z be the z transformable edges of T with mx < mx + 1 for all x. Let tx be the path length from the vertex vm x to the corresplnding pendant vertex decided by the transformable edge vmx vhx of T. Define a labeling f : V (S(T )) → N by

f (vi ) = k + (q − 1)d + (i − 1)d, f (vi,j ) = (i − 1)d,

if i=1, 2...n

if j 6= i + 1

f (vi,j ) = id, if j = i + 1 and i = mc , mc + 1, ...mc + tc − 1, c = 1, 2, ...z f (vi,j ) = (i − 1)d, if j = i + 1 and i 6= mc , mc + 1, ...mc + tc − 1, c = 1, 2, ...z

where k and d are positive integers and 2q is the number of edges of S(T). Clearly f is a (k+(q-1)d, d)-arithmetic labeling of S(T).  This is a wonderfull result for arithmetic labelling of subdivision of a Tp -tree. As an example, we may use Theorem 5.2.1 and obtain the (14, 1)-arithmetic labeling of subdivision of a Tp -tree and it is shown in the following Figure 5.3.

37

Fig 5.3 ? ?

? ? ? ? ? ? ? ?

38

CLASSES OF ARIT HMET IC GRAPHS In this chapter we shall give special classes of graphs that are amenable to arithmetic numbering 6.1 Arithmetic Numbering of Bipartite Graphs We know that every connected, balanced, bipartite graph is arithmetic. We shall offer wider ranges of values of k and d than provided by Theorem 3.3.11 for certain special classes balanced bipartite graphs. Henceforth, unless specified otherwise, any bipartite graph G with bipartition {A,B} will be assumed to be accompanied with a labeling of it’s vertices given by A = {u1 , u2 , ...ua } and B = {v1 , v2 , ...vb } where a ≤ b. THEOREM 6.1.1 For any positive integer k and d and for any partition of k into two parts k1 and k2 with 0 ≤ k1 < k2 , the star K1,b has a (k,d)-arithmetic numbering f such that k1 , k2 ∈ f (K1,b ). Proof Let A={u } (i.e., u1 = u) and f : A ∪ B −→ N be a map defined by f (u) = k1 f (vi ) = k2 + (i − 1)d, 1 ≤ i ≤ b

) (13)

It is easily seen that f is a required (k, d)-arithmetic numbering of K1,b .  Thus, K1,b is “arbitrarily arithmetic” in the sense that it is (k, d)-arithmetic for all values of k and d. According to Theorem 3.1.6, there are no other connected bipartite graphs with this property. By Theorem 3.1.7, there is no Eulerian graph with this property. It would be interesting to know if there is any graph different from K1,b that is arbitrarily arithmetic. THEOREM 6.1.2 For any positive integer k and d and for any partition of k into two parts k1 and k2 with 0 ≤ k1 < k2 , the complete bipartite graph Ka,b , 2 ≤ a ≤ b has a (k, d)-arithmetic numbering f such that k1 , k2 ∈ f (G) where either C1: d | (k2 − k1 ) or C2: k2 − k1 = rd for some r ≥ a. Proof Consider the map f : A ∪ B −→ N defined by ) f (ui ) = k1 + (i − 1)d, 1 ≤ i ≤ a (14) f (vj ) = k2 + (j − 1)ad, 1 ≤ j ≤ b 39

Then it is easily seen that f + (Ka,b ) = {k, k + d, k + 2d, ...k + (q − 1)d} where q=ab. So, it remains to see that f is bijective in the respective cases C1 and C2. Suppose first that C1 holds and f (ui ) = f (vj ) for some i, j. Then by equation (14) we get k1 + (i − 1)d = k2 + (j − 1)ad ⇒ k2 − k1 = [(i − 1) − (j − 1)a]d. The last eqality contradicts C1. Thus f must be injective and hence is a required (k,d)arithmetic numbering of Ka,b in this case. On other hand, if under C2 we have f (ui ) = f (vj ) we get

rd = k2 − k1 = [(i − 1) − (j − 1)a]d ⇒ (i − 1) − (j − 1)a = r ≥ a ⇒ (i − 1) ≥ ja which is a contradiction as 1 ≤ j ≤ b. Hence f is a (k, d)-arithmetic numbering of Ka,b when C2 holds.  One can in fact prove that each of C1 and C2 is also neccessary for Ka,b , 2 ≤ a ≤ b, to have a (k, d)-arithmetic numbering f with k1 , k2 ∈ f (Ka,b ) and the proof, which is rather tedious, essentially makes use of the following general property of arithmetic numbers of Ka,b . LEMMA 6.1.3 For any f ∈ Ak,d (Ka,b ) either {f + (ui vj ), 1 ≤ j ≤ b} = {k +((i−1)b+(j −1))d, 1 ≤ j ≤ b} for each i, 1 ≤ i ≤ a or {f + (ui vj ), 1 ≤ i ≤ a} = {k + ((j − 1)a + (i − 1))d, 1 ≤ i ≤ a} for each j, 1 ≤ j ≤ b.  Example 6.1.4 Let (a, b, k, d)=(3, 4, 10, 2). Then the partitions (k1 , k2 ) of k with 0 ≤ k1 < k2 are (0, 10), (1, 9), (2, 8), (3, 7), (4, 6). Then d | (k2 − k1 ) for each of these partitions (k1 , K2 ), but only for the first three of these do we have r ≥ 3 = a, k2 − k1 = 2r. The corresponding (k,d)-arithmetic numbering of K3,4 for these three cases are shown in Figure 6.1 As another example, (a, b, k, d)=(3, 4, 12, 3), then the partitions (k1 , k2 ) of k with 0 ≤ k1 < k2 are (0, 12), (1, 11), (2, 10), (3, 9), (4, 8), (5, 7). Then only (0, 12) satisfies C2 and hence when (k1 , k2 ) is any of these partitions of k, the graph K3,4 has a (k,d)arithmetic numbering f with k1 , k2 ∈ f (K3,4 ). The partition (3,9) does not satisfy either C1 or C2, hence K3,4 has no (k, d)-arithmetic numbering f for which 3, 9 ∈ f (K3,4 ).

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Fig 6.1 Definition 6.1.5 Caterpillars are trees for which the removal of their end vertices results in a simple path. We denote by Ca,b a caterpillar with bipartition {A, B}, where A = {u1 , u2 , ...ua } and B = {v1 , v2 , ...vb } and a ≤ b. We note here that given a, b, Ca,b is not necessarily unique. It is well known that every caterpillar has a balanced numbering, as displayed, for instance, on the plane representation of particular Ca,b shown in Figure 6.2.

Balance numbering the plane representation of caterpillar Ca,b , where p=a+b Fig 6.2 Therefore, by Theorem 3.3.11, all caterpillars are arithmetic. The following theorem shows that they admit a (k, d)-arithmetic numberings f with k1 , k2 ∈ f (Ca,b ) when (k1 , k2 ), 0 ≤ k1 < k2 , is a partition of k satisfying C1 or C2 of Theorem 6.1.2.

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THEOREM 6.1.6 For any positive integer k and d and for any partition of k into two parts k1 and k2 with 0 ≤ k1 < k2 , Ca,b , 1 ≤ a < b, has a (k, d)-arithmetic numbering f with k1 , k2 ∈ f (Ca,b ) if k1 , k2 satisfies either C1 or C2 of Theorem 6.1.2. Proof The case a=1 is that of the star K1,b =C1,b , and by Theorem 6.1.1, for any partition of k into two parts k1 and k2 with 0 ≤ k1 < k2 the map f of (13) is a required (k, d)-arithmetic numbering of C1,b . Hence, we assume that 2 ≤ a < b. Also we shall assume throughout this proof that the vertices of Ca,b are labeled from top to bottom, as shown in plane representation of Ca,b in Figure 6.2. Now consider the map f : A ∪ B −→ N defined by f (ui ) = k1 + (i − 1)d, 1 ≤ i ≤ a f (vj ) = k2 + (j − 1)d, 1 ≤ j ≤ b

) (15)

Then

f + (Ca,b ) = {f + (ui vj ) = k + (i + j − 2)d, 1 ≤ i ≤ a, 1 ≤ j ≤ b} = {k, k + d, ...k + (a + b − 2)d}. Hence f (ui ) = f (vj ) yields k2 − k1 = (i − j)d

(16)

Therefore, if d | (k2 − k1 ), then (16) is a contradiction, and on the other hand, if k2 − k1 = rd, r ≥ a, then (16) yields

rd = k2 − k1 = (i − j)d ⇒ i−j =r ≥a ⇒ i≥a+j a contradiction to the fact that 1 ≤ i ≤ a and 1 ≤ j ≤ b. Thus f must be a required (k, d)-arithmetic numbering of Ca,b .  However, unlike Ka,b , the graph Ca,b may have a (k,d)-arithmetic numbering f with k1 , k2 ∈ f (Ca,b ), for a partition (k1 , k2 ) of k, 0 ≤ k1 < k2 , not satisfying either C1 or C2 of Theorem 6.1.2. Figure 6.3 displays two such instance.

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Fig 6.3 It seems a hard problem to determine in general the full range of values of k and d and possible partitions (k1 , k2 ) of k, 0 ≤ k1 < k2 , with k2 − k1 = rd and r < a such that Ca,b , 2 ≤ a ≤ b, has a (k, d)-arithmetic numbering f for which k1 , k2 ∈ f (Ca,b ). In particular, if (a, b, k, d)=(a, b, a, 1) then for the partition (0, a) of a, since 1 | (a − 0) and r=a, we see that Ca,b a ≤ b, has an (a, 1)-arithmetic numbering f with 0, a ∈ f (Ca,b ). But then since f (Ca,b ) = {0, 1, 2, ..a + b − 1} and f + (Ca,b ) = {a, a + 1, ..., 2a + b − 2} we obtain Corollary 6.1.7. COROLLARY 6.1.7 Every caterpillar is sequential.  Conjecture 1 For any quadruple (a, b, k, d) of positive integers and a partition (k1 , k2 ), 0 ≤ k1 < k2 of k, satisfying either C1 or C2 of Theorem 6.1.2, any balanced bipartite graph G with {A, B}, | A |= a ≤ b =| B |, has a (k, d)-arithmetic numbering f with k1 , k2 ∈ f (G). 6.2 Arithmetic Numbering of Nonbipartite Graphs We shall next treat some classes of nonbipartite graphs Consider the complete tripartite graph Ka,b,1 = Ka,b + K1 with tripartition {A, B, C} where A = {u1 , u2 , ..., ua }, B = {v1 , v2 , ..., vb } and C = {w}. THEOREM 6.2.1 The graph Ka,b,1 , a ≤ b, is (k+2r, k)-arithmetic for all integers k ≥ 1 and r ≥ 0. Proof Define a map f : A ∪ B ∪ C −→ N by f (w) = r f (vj ) = kj + r,

1≤j≤b

f (ui ) = (b + 1)ki + r, 1 ≤ i ≤ a 43

can be easily verified to be a required (k+2r,k)-arithmetic numbering of Ka,b,1 .  Example 6.2.2

Fig 6.4 The only possible (k, d)-arithmetic numbering of K3 and K4 are displayed in Figure 6.5, 0 0 0 where k = k + 2r and d = k for integers k ≥ 1 and r ≥ 0.

Fig 6.5 For the class of complete graphs in general we have the following conjecture to propose: Conjecture 2 For any integer n ≥ 5, Kn is not arithmetic. For any cycle Cn = {u1 , u2 , ..., un , u1 }, n ≥ 4, the following four theorems hold: THEOREM 6.2.3 For any positive integer k and d for any partition (k1 , k2 ) of k with 0 ≤ k1 < k2 satisfying C1 and C2 of Theorem 6.1.2 there exists a (k, d)-arithmetic numbering f of C4t , t ≥ 1, such that k1 , k2 ∈ f (C4t ). 44

Proof Define f : V (C4t ) −→ N by  i−1   k1 + ( 2 )d, if i is odd f (ui ) = k2 + ( i−2 )d, if i is even and 2 ≤ i ≤ 2t 2   i k2 + ( 2 )t, if i is even and 2t + 2 ≤ i ≤ 4t

(17)

One may then easily verify that f turns out to be a (k, d)-arithmetic numbering of C4t where C1 and C2 of Theorem 6.1.2 is satisfied.  Figure 6.6 shows a (1, 2)-arithmetic numbering f of C8 (under C1) and an (8,2)-arithmetic numbering of C8 (under C2).

Fig 6.6 THEOREM 6.2.4 For any nonnegative integer r and positive integer d, C4t+1 is (2dt+2r, d)-arithmetic. Proof Under the hypotheses the map f : V (C4t+1 ) −→ N defined by ( f (ui ) =

)d, if i is odd r + ( i−1 2 1 2dt + r + 2 id, if i is even

can be easily verified to be a required arithmetic numbering of C4t+1 .  Figure 6.7 displays such an arithmetic numbering with d=2 and r=0 for C9 .

45

(18)

Fig 6.7 THEOREM 6.2.5 C4t+1 is not arithmetic. Proof Suppose that C4t+2 = {u1 , u2 , ..., u4t+2 , u1 } has a (k,d)-arithmetic numbering f. Let then f (ui ) = k + (i − 1)d for each i, 1 ≤ i ≤ 4t + 2 where ui ui+1 ∈ E(C4t+2 ) with indices reduced modulo 4t+2. Without loss of generality, we may assumed that x1 + x2 = k. Then Theorem 2.1.5 we get 2(x1 + x2 ) + 2

4t+2 X

xi = (4t + 2)k + (2t + 1)(4t + 1)d

i=3

⇒ 2k + 2{(x3 + x4 ) + (x5 + x6 ) + .... + (x4t+1 + x4t+2 )} = (4t + 2)k + (2t + 1)(4t + 1)d ⇒ 2k + 2(2tk + md) = 2k + 4tk + (2t + 1)(4t + 1)d where m is a positive integer. Hence 2md = (2t + 1)(4t + 1)d ⇒ 2m = (2t + 1)(4t + 1)d This is a preposterous identity sice both sides are positive integers.  THEOREM 6.2.6 For any nonnegative integer r and positive integer d, C4t+3 is ((2t+1)d+2r, d)-arithmetic. Proof Under the hypotheses, the map f : V (C4t+3 ) −→ N defined by

46

( f (ui ) =

r + ( i−1 )d, if i is odd 2 1 (2t + 1)d + r + 2 id, if i is even

(19)

is a required arithmetic numbering of C4t+3 . This completes the proof.  This numbering is diplayed on C7 in Figure 6.8 by taking d=2 and r=0.

Fig 6.8 We observe, invoking Theorem 2.1.5, that for an odd cycle to be (k, d)-arithmetic it is necessary that k and d to be parity. This prompts us to the following: Conjecture 3 (1) If C4t+1 is (k, d)-arithmetic then k=2dt+2r for some integer r ≥ 0. (2) If C4t+3 is (k, d)-arithmetic then k=(2t+1)d+2r for some integer r ≥ 0. ? ?

? ? ? ? ? ? ? ?

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REFERENCES [1] Acharya, B. D and Hegde, S. M, Arithmetic graphs, J.Graph Theory, 14(3), 1989, 275-299. [2] Balakrishnan, R and Ranganathan, K, A Text Book of Graph Theory, Springer, Newyork, 1996. [3] Chartrand, Gray and Lesniak, Linda, Graphs and Digraphs, Chapman and Hall, London, 1996. [4] Clark, John and Holton, D. A, A First Look At Graph Theory, Allied Publishers Ltd, Singapore, 1995. [5] Gallian, Joseph. A, A Dynamic Survey of Graph Labeling, Electron. J. Combinatorics, 10, 2007, DS6. [6] Hegde, S.M and Shetty, Sudhakar, On Arithmetic Graphs, Indian J. pure appl.Math., 33(8), 2002, 1275-1283.

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