Odd harmonious labeling of Sn (m, r) graph

Abstract

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.