Some Results on Vertex Equitable Labeling

Abstract

Let G be a graph with p vertices and q edges and let A= vertex labeling is said to be a vertex equitable labeling of G if it induces an edge labeling given by such that and , where is the number of vertices v with for A graph G is said to be a vertex equitable graph if it admits vertex equitable labeling. In this paper, we establish the vertex equitable labeling of a Tp-tree, where T is a Tp-tree with even number of vertices, bistar the caterpillar and crown

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Jeyanthi, P. and Maheswari, A. (2012) Some Results on Vertex Equitable Labeling. Open Journal of Discrete Mathematics, 2, 51-57. doi: 10.4236/ojdm.2012.22009.

1. Introduction

All graphs considered here are simple, finite, connected and undirected. We follow the basic notations and terminologies of graph theory as in [1]. The symbols and denote the vertex set and the edge set of a graph G. Let be a graph with vertices and edges. A labeling f of a graph G is a mapping that assigns 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 for each edge xy which is called as weight of the edge xy.

A difference labeling f of a graph G is said to be k-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 [3]. A brief summary of definitions which are useful for the present study is given below.

Definition 1.1 [4] Let T be a tree and u0 and be two adjacent vertices in T. Let u and v be two pendant vertices of T such that the length of the path u0-u is equal to the length of the path -v. If the edge 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, for short) and the edge is called 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 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 Tp tree and a sequence of two ept’s reducing it to a path are illustrated in Figure 1.

Definition 1.2 The corona of the graphs G1 and G2 is obtained by taking one copy of G1 (with p vertices) and p copies of G2 and then joining the vertex of G1 to every vertex of the copy of G2.

Definition 1.3 Caterpillar is a tree with the property that the removal of its pendant vertices leaves a path.

Definition 1.4 The square graph G2 of a graph G has the vertex set with adjacent in G2 whenever in G.

denotes the smallest integer greater than or equal to x.

The concept of mean labeling was introduced by S. Somasundaram and R. Ponraj in [5] and further studied in [6-8]. A. Lourdusamy and M. Seenivasan introduced a vertex equitable labeling in [9]. In a vertex equitable labeling we use the labels for the vertices,

Figure 1. A Tp-tree and a sequence of two ept’s reducing it to a path.

the number of times the different vertex labels appear cannot differ by more than one. The induced edge labels are defined as the sum of the incident vertex labels. They proved that the graphs like path, bistar combs bipartite complete friendship graph for quadrilateral snake,

if and only if ladder graph arbitrary super division of a path and cycle with or are vertex equitable. Also they proved that the graph if Eulerian graph with n edges where or the wheel the complete graph if and triangular cactus with q edges where or 6 or 9 are not vertex equitable. Moreover they proved that if G is a graph with p vertices and q edges, q is even and then G is not vertex equitable.

Definition 1.5 [9] Suppose G is a graph with p vertices and q edges. Let A vertex labeling induces an edge labeling defined by for all edges uv. For let be the number of vertices v with A graph G is vertex equitable if there exists a vertex labeling f such that for all a and b in A,

and the induced edge labels are

P. Jeyanthi and A. Maheswari proved in [10,11] that tadpoles, Cm Cn, armed crowns, [Pm;] and, , the graphs obtained by duplicating an arbitrary vertex and an arbitrary edge of a cycle Cn, total graph of Pn, splitting graph of Pn and fusion of two edges of a cycle Care vertex equitable graphs. In this paper, we establish the vertex equitable labeling of a Tp-tree, where T is a Tp-tree with even number of vertices, the bistar the caterpillar and the crown

2. Main Results

Theorem 2.1 Let and be any two vertex equitable graphs with equitable labeling f and g respectively. Let u and v be the vertices of G1 and G2 respectively such that and Then the graph obtained from G1 and G2 by identifying the vertices u and v is a vertex equitable graph.

Proof. Clearly has edges and vertices. Let

Define

by for and for Clearly,

Therefore, and the labels of the edges of the copy of G1 are and the labels of the edges of the copy of G2 are Hence, is a vertex equitable graph.

Theorem 2.2 Let and be any two vertex equitable graphs with equitable labeling f and g respectively. Let u and v be the vertices of G1 and G2 respectively such that and Then the graph G obtained by joining u and v by an edge is vertex equitable.

Proof. Clearly G has edges and vertices. Let

Define

by if if The labels of the edges of the copy of G1 are and the labels of the edges of the copy of G2 are and

Hence, G is a vertex equitable graph.

Theorem 2.3 Every Tp-tree is a vertex equitable graph.

Proof. Let T be a Tp-tree with n vertices. By the definition of a transformed tree there exists a parallel transformation P of T such that for the path we have 1), 2) where is the set of edges deleted from T and is the set of edges newly added through the sequence of the epts P used to arrive the path Clearly, and have the same number of edges.

Now denote the vertices of successively as starting from one pendant vertex of right up to the other.

For define the labeling f as

Then f is a vertex equitable labeling of the path

Let be any edge of T with and be the ept that deletes this edge and add the edge where t is the distance of from and also the distance of from Let P be a parallel transformation of T that contains as one of the constituent epts.

Since is an edge of the path it follows that which implies Therefore and are of opposite parity.

The induced label of the edge is given by

Now

Therefore, we have and hence f is a vertex equitable labeling of the Tp-tree T.

An example for the vertex equitable labeling of a Tp- tree with 12 vertices is given in Figure 2.

Theorem 2.4 Let T be a Tp-tee with even number of vertices. Then the graph is a vertex equitable graph for all

Proof. Let T be a Tp-tree of even order m and the vertex set Let be the pendant vertices joined with by an edge. Then

By the definition of a Tp-tree, there exists a parallel transformation P of T such that for the path we have 1), 2) where is the set of edges deleted from T and is the set of edges newly added through the sequence of the epts P used to arrive the path Clearly, and have the same number of edges.

Now denote the vertices of successively as starting from one pendant vertex of right up to the other. The labeling f defined by

Figure 2. Vertex equitable labeling of a Tp-tree with 12 vertices.

is a vertex equitable labeling graph.

Let be any edge of T with let be the ept that deletes this edge and adds the edge where t is the distance of from and also the distance of from Let P be a parallel transformation of T that contains as one of the constituent epts.

Since is an edge in the path it follows that which implies Therefore i and j are of opposite parity.

The induced label of the edge is given by

Therefore, we have and thus f is a vertex equitable labeling of

An example for the vertex equitable labeling of where T is a Tp-tree with 12 vertices is shown in Figure 3.

Let be a graph obtained from by attaching n pendant edges at one vertex and pendant edges at the other vertex.

Theorem 2.5 The bistar is a vertex equitable graph.

Proof. Let and and be the vertices adjacent to u and v respectively. Now, has edges and vertices. Define

by if and if Then f is a vertex equitable labeling of

Theorem 2.6 Let and

Then is a vertex equitable graph.

Figure 3. Vertex equitable labeling of.

Proof. By Theorem 2.5, is a vertex equitable graph. Let be the corresponding vertex equitable labeling of Let Since Consider the graphs and The number of edges of the graph is.

Now,

Therefore, by Theorem 2.1, is a vertex equitable graph. Let be the corresponding vertex equitable labeling of Again the number of edges of is even.

Now take Hence Also

Therefore, by Theorem 2.1, is a vertex equitable graph and the number of edges is even. Proceeding like this, at the step we get is a vertex equitable graph where

Let be the corresponding vertex equitable labeling of Take

Clearly Now,

Therefore, is a vertex equitable graph.

An example for the vertex equitable labeling of if n is odd is given in Figure 4.

An example for the vertex equitable labeling of if n is even is given in Figure 5.

Figure 4. Vertex equitable labeling of S (4, 6, 9, 7 + 1).

Figure 5. Vertex equitable labeling of S (5, 7, 9, 10, 2 + 1).

Theorem 2.7 The crown is a vertex equitable graph.

Proof: Let be the vertices of the cycle Cn and let vi be the vertex adjacent to ui for Then the vertex set and the edge set. Define for the following cases:

Case 1.

Case 2.

Case 3.

Figure 6. Vertex equitable labeling of.

Case 4.

In all the above cases, f is a vertex equitable labeling. Hence is a vertex equitable graph.

An example for the vertex equitable labeling of is shown in Figure 6.

Theorem 2.8 The graph is a vertex equitable graph.

Proof. Let be the path Clearly, has n vertices and edges. Define

by Evidently, is a vertex equitable graph.

Conflicts of Interest

The authors declare no conflicts of interest.

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