Even vertex odd mean labeling of uniform theta graphs

Let G be a graph with q edges. A graph G is called even vertex odd mean graph if there exist an injective labeling f : V ( G ) → { 0 , 2 , 4 , 6 , ..., 2 q } with an induced edge labeling f ∗ : E ( G ) → { 1 , 3 , 5 , ..., 2 q − 1 } such that for each edge uv , f ∗ ( uv ) = f ( u )+ f ( v ) 2 is bijection. In this paper we obtain su ﬃ cient conditions for certain uniform theta graphs to be even vertex odd mean graphs.


Introduction
Let G = (V, E) be a simple, connected and undirected graph with vertex set V (G) and edge set E(G).The notions and terminology that exist in this paper can be found in Harary [6].In [9] Rajan et al. defined a generalized theta graph θ(m 1 , m 2 , ..., m n ) is the graph obtained by taking two different isolated vertices u and v (called the end vertices of θ) and attaching them by n internal disjoint paths of length greater than one, where m i , 1 ≤ i ≤ n denote the order of internal vertices of ith path.The two isolated vertices u and v are called north pole (N ) and south pole (S) respectively.A generalized theta graph is called uniform theta graph if all internal disjoint paths have the same order i.e. m i = m for all 1 ≤ i ≤ n.A uniform theta graph with n internal disjoint paths, all have the same order m is denoted by θ(n; m).In Figure 1 we show the representation of θ(3; 5).R. Vasuki et al. defined the even vertex odd mean labeling on graph G = (V, E) as an injective vertex labeling f from V (G) to {0, 2, ..., 2q} such that the induced edge labeling f * is defined by for every edge uv ∈ G.The resulting edge labels are odd distinct integers from {1, 3, 5, ..., 2q − 1} [10].A graph that admits an even vertex odd mean labeling is said to be an even vertex odd mean graph [ [1], [2], [3], [4], [7], [8], [11]].Labeled graphs are used in coding theory, cryptography, mathematical modelling, x-ray, crystallography, and determining optimal circuit layouts, also graph theory is important in many areas of computer science study, including networking, database management systems, and artificial intelligence [12].Gallian provides a current survey of numerous graph labeling challenges as well as a comprehensive bibliography [5].

Main results
Theorem 1.If n is odd and m ≥ 2, then θ(n; m) is an even vertex odd mean graph.

Case(i).
When m is odd.Let P i m denote the ith internal path of θ(n; m), hence the vertices P 1 m and P 2 m are labeled as follows: Now if the vertices of the internal path P i−2 m are labeled by f , hence the vertices of the internal path P i m are labeled as follows: Clearly, the induced edge labeling f * is obtained as follows: Let E u and E v be the set of edges joining between the end points of n disjoint paths and the two isolated vertices u and v respectively.Then we denote the set of edge labels of E u and E v by f * (E u ), f * (E v ) respectively.Thus, it is observed that: Also let E(P i m ) be the set of all edges of the ith internal path of θ(n; m).We denote the set of edge labels of E(P i m ) by f * (E(P i m )).So it is easy to see that:

M. Basher
It follows that the induced edge labels of E(P i m ) are given by: Hence, f is an even vertex odd mean labeling of θ(n; m).
Case(ii).When m is even.
Here the vertices of 1st and ( n+3 2 )-th internal paths label as follows: 1 ≤ j ≤ m and j is even Also the remaining vertices of other internal paths label as follows: Therefore, we can compute the induced edge labels as follows: Also the set of edge labels of E(P 1 m ) and E(P ) are given as follows: Clearly, the edge labels of E(P i m ) are given by: Hence, f is an even vertex odd mean labeling of θ(n; m). 2 Example 2.1.The even vertex odd mean labeling f of θ(7; 5) and θ(9; 6) are given in Figure 2 and Figure 3 respectively.

Case(i).
When n is even and m is odd.In this case the vertices of the internal paths u ij (1 ≤ i ≤ n, 1 ≤ j ≤ m) label as the following, first we label the vertices of the paths P 1 m , P 2 m by: 1 ≤ j ≤ m and j is even.
are labeled, then the vertices of the path P i m are labeled as follows: 1 j ≤ m and j is even.
According to Theorem 1. we can compute the induced edge labels as follows: Suppose the edge of the path P i−2 m are obtained, hence for 1 ≤ i ≤ n 2 and n 2 + 3 ≤ i ≤ n the edge labels of P i m are labeled as follows: For i = n 2 + 1, n 2 + 2 the edge of the two internal paths E(P n 2 +1 m ) and ) are labeled by: f Therefore, f is an even vertex odd mean labeling of θ(n; m).

Case(ii).
When n = 4 and m is even.
Here the vertices of the internal paths P 1 m , P 2 m , P 3 m and P 4 m are labeled as follows: 1 ≤ j ≤ m and j is odd 6m + 2j + 6, 1 ≤ j ≤ m and j is even.
1 ≤ j ≤ m and j is even.
1 ≤ j ≤ m and j is even.
Clearly, the edge labels of the four internal paths are given as follows: Thus, f is an even vertex odd mean labeling of θ(n; m).For other even values of m the problem becomes NP-hard problems.2 Example 2.2.The even vertex odd mean labeling f of θ(8; 7) and θ(4; 12) are given in Figure 4 and Figure 5 respectively.

Figure 2 . 3 :Figure 2 . 4 :
Figure 2.3: An even vertex odd mean labeling of θ(8; 7) If n is even and m is odd or n = 4 and m is even, then θ(n; m) is an even vertex odd mean graph.