Some Results on Vertex Equitable Labeling

Let G be a graph with p vertices and q edges and let = 0,1, 2, , . 2 q A              A vertex labeling   : f V G A  i e a vertex equitable labeling of G if it induces an edge labeling s said to b given by      * =  f uv f u f v  * f such that     1 v a v b   and     * = 1,2,3, , f f f E q  , where   f v a is the ber of verti num ces v with   = f v a for . a A  to be a h if it adm tex equitable labeling. In this paper, we the equitable labeling of a Tp-tree, A graph G is said r abl vertex vertex equitable grap its ve est ish n T K  where T is a Tp-tree with even number of vertices, bistar   , 1 , B n n  the caterpillar   1 2 , , , S x x x  and crown 2 1 . n n C K n ,  P eywords: Vertex Equitable Labeling; Vertex Equitable Graph

e a vertex equitable labeling of G if it induces an edge labeling s said to b given by , where where T is a T p -tree with even number of vertices, bistar   , 1 , B n n  the caterpillar   edges.A labeling f G is a mappin ns elements of a graph to the set of numbers (usually to positive or non-negative integers).If the domain of the mapping is the set of vertices (the set of edges) then we call the labeling vertex labeling (edge labeling).The labels of the vertices induce labels of the edges.There are several types of labeling.A detailed survey of graph labeling can be found in [2].A vertex labeling f is said to be difference labeling if it induces the label     E G g that assig of a graph f x f y  for each edge xy which is called as weigh .A difference labeling t of the edge xy f of a graph G is said to be ktree and u 0 and be tw to the length of the path -v.If the edge is desuch a transformation of lled an elemen ry parallel transformation (or a pt, for short) e edge is called transformable edge.by a sequence of ept can be reduce a path, then T is called a T p tree (transformed tree) and such sequence is 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 T p tree and a sequence of two ept's reducing it to a path are illustrated in Figure  [3].A brief summary of definitions which are useful for the present study is given below.Definition 1.1 [4] Let T be a G G  one copy d then join copy of r is a tree with the p denotes the smallest integer r than or equal to The concept of mean labeling was introduced by S. Somasundaram and R. Ponraj in [5] and further studied in [6][7][8].A. Lourdusamy and M. Seenivasan introduced a vertex equitable labeling in [9].In a vertex equitable labeling we use the labels greate for the vertices, ,  

f v a x la-A graph G is vertex equitable if there ex s a verte beling f such that for all a and b in
and the induced e e labels are 1, 2,3, , .q  P. Je i and A. M ri proved in [10,11] and the crown 2 1 , .
, G p q ble labelin vertices of G 1 be any two vertex equitable gr ith equita g f and g respectively.Let and G 2 respectively such that aphs w u and v be the Then the graph Clearly and the labels of the edges of the copy of G 1 are 1, 2, , 2n  and the labels of the edges of the copy of , 2 1 p n ith et u and v be th able labelin e vertices of G 1 a 1 any two vertex and g respectively.L G 2 respectively suc  = f u n and   = 0. v Then the graph G obtained by joining u and v by an edge is vertex equitable.
Proof.Clearly G has 2 2 2 edge and 1 bels of the edges of the copy of G 1 he labels of the edges of the ce, G is a verte table graph.Theorem 2. 3 T p -tree is quitable graph.
Proof.Let T be a T p -tree with n vertices.By the definition of a transformed tree there exists a parallel ansformation P of T such that for the path we have 1) 2)

Hen
x equi Every a vertex e tr   define the labeling f as   Then f is a vertex equitable labeling of the path   it follows that = 2

Therefore
e of opposite parity is given by i and j ar .The induced label of the edge i j v v f is a vertex equitable labelin An example for the vertex equitable labeling of a T ptree with 12 vertices is given in Figure 2.
Theorem 2. 4 Let T be a T -tee with even number of vertices.Then the graph j t  g of the T p -tree T. p n T K  is a vertex equitable f.L order m and r-graph for all 1. n  Proo et T be a T p -tree of even the ve tex set     , , , By the def T p -tree, there exis transformat on P of T such that for the path   where dge

  P T ndant v
Now denote the vertices of successively as starting from one pe ertex of

 
Therefore, we have An example for the vertex equitable labeling of u v and and   Since is an edge in the path it follows that which implies .Therefore i and j are of opposite parity.

K
All graphs considere and undirected.We follow the basic notations and terminologies of graph theory as in[1].The symbols   V G and   E G denote the vertex set and the edge se graph et vertices in T. Let u and v be two pe ant vertices of T such that the length of the path u 0 -u is equal 0 leted from T and u and v are joined by an edge uv, then v up to the other.The labeling ned by f defi

Figure 2 .
Figure 2. Vertex equitable labeling of a T p -tree with 12 vertices.
even is given in Figure5.


Figure 6.Vertex eq i The square graph G 2 of a graph G has the vertex set equitable if for each weight induced by f on the edges of G appears exactly k times.If a graph G has a k-equitable labeling then G is said to be k-equitable.Equitable labeling of graphs was introduced by Bloom and Ruiz in

two ept's reducing it to a p Figu T p -tree and a sequenc the number of times the different vertex labels appear re 1. A e o ath.
n that tadpoles, C m  C n , armed crowns, [P m ; n graph of P n , splitting graph of P n and fusion of two edges of a cycle C n are vert In this paper, we esta vertex equitable labeling o T p -tree, n T  ere is a p -tree with even number o ces, the bistar

table graph
4, 6,9, 7 1 S  if n odd is given in