Odd harmonious labeling of Sn (m, r) graph

A graph labeling is an assignment of integers to vertices or edges of a graph subject to certain conditions. There are various kinds of graph labeling, one of them is an odd harmonious labeling. An odd harmonious labeling f of a graph G on q edges is an injective function f from the set of vertices of G to the set {0,1,2,…,2q - 1} such that the induced function f*, where f* (uv) = f(u) + f(v) for every edge uv of G, is a bijection from the set of edges of G to {1,3,5,…,2q - 1}. A graph is said to be odd harmonious if it admits an odd harmonious labeling. A graph Sn(m, r) is a graph formed from r stars, each of which has n + 1 vertices, and every center of the star is joined to one new vertex v 0 by a path of length m. In this paper we show that the graph Sn (m, r), m ≥ 2, 1≤ r≤3, is odd harmonious.


Introduction
Graph theory is a branch of mathematics that many people work on it. In this paper, we only cosider finite and simple graph. We follow that of Diestel [1] for most part of notation and terminology. Here, G denotes a graph, V(G) and E(G) denote the vertex set and the edge set of G, respectively. When A is a set, then cardinality of A is denoted by A . Hence, ) (G V and ) (G E are the number of vertices and the number of edges of G, respectively.
One of the topics in graph theory is graph labeling. Graph labeling was first introduced in the mid 1960's. The definition of graph labeling can be found in Gallian [3]; it is an assignment of integers to the vertices or edges subject to certain conditions. More than 2800 papers on graph labelings have been published [3].
Jeyanthi and Philo [4,5] studied odd harmonious labeling of certain graphs, including subdivided shell graphs. Further, Jeyanthi, et al. [6] studied odd harmonious labeling of super subdivision graphs. and Jeyanti, et al. [7] studied odd harmonious labeling of grid graphs. Liang and Bai [8] have studied some classes of graphs, including path, cycles, complete graphs, complete k-partite graphs, and windmill graphs. Selvaraju, et al. [9] studied odd harmonious labeling of some path related graphs. Vaidya and Shah [10] studied odd harmonious labeling of some graphs including shadow graph and splitting graph. Recently, Febriana and Sugeng [2] studied odd harmonious labeling on squid graph and double squid graph. Let n, m and r, be positive integers. A graph ( , ) is a graph formed from stars, each of which has + 1 vertices, and every center of the star is joined to one new vertex 0 by a path of length . From the above discussion, we can see that the odd harmonious labeling of ( , ) has not been studied. In this paper we study the odd harmonious labeling of ( , ) for the case ≥ 2, 1 ≤ ≤ 3, and show that ( , ) is odd harmonious. A special case, when = = 1, graph ( , ) is a path. Liang and Bai [8] say that paths are odd harmonious.

Proof:
when is odd, 1 ≤ ≤ 2( − 1) + , when is even, 1 ≤ ≤ ( 0 ) = { + 2 − 1, when is odd + 1, when is even  3 We show will that all the values of f are different. It is easy to see that the odd values of f are different, ( 1 ) = when i is odd, and 1 ≤ ≤ ; and ( 0 ) = + 1 when m is even.. The even values of f are ( 1,  Thus we can see that all values of * ( ) are odd and different, * is injective. This completes the proof that graph ( , 2) is odd harmonious.
The third result is for the case = 3.  This completes the proof that all values of * ( ) are odd and different, * is injective, and hence graph ( , 3) is odd harmonious. Then, the odd harmonious labeling to graph ( , 3) for positive integers applies to ≥ 2. It is because when = 1 or when graph (1,3) there is the same vertex label so that is not an injective function. Therefore, the graph (1,3) for positive integers cannot be labeled with odd harmonious labeling. So it is proven that odd harmonious labeling to graph ( , 3) for positive integers applies when ≥ 2. This completes the proof that graph ( , 3) is odd harmonious when ≥ 2. For example, graphs 6 (7,1), 7 (4,2), and 6 (5,3), are shown in Figure 4, Figure 5, and Figure 6, respectively.    Figure 6. Odd harmonious labeling to 6 (5,3) graph.

Conclusion
We have shown that graph ( , ) for 1 ≤ ≤ 3, is harmonious. When = = 1, ( , ) is a path. So, this result is more general than the result which says that paths are harmonious, that can be found in Liang and Bai [8]. There is still a problem for the general case. For next study, one can prove (or disprove for some cases) that ( , ) is odd harmonious, for > 3.