On odd harmonious labeling of m -shadow of cycle, gear with pendant and Shuriken graphs

Abstract

A graph G = (V(G), E(G)) with p vertices and q edges is called (p, q)-graph. An injection f is said to be odd harmonious labeling of a (p, q)-graph G if there is an injective function f from a set of vertices V(G) to a set {0, 1, 2, .., 2q - 1} such that the induced function f∗ from a set of edges E(G) to a set of odd number {1, 3, 5, . 2q - 1} defined by f ∗ (uv)=f (u)+f (v) is a bijection. A graph G is said to be odd harmonious if there exists an odd harmonious labeling for G. In this paper we proved that several product graphs, such as m-shadow of cycle Dm (Cn), gear with pendant graphs and Shuriken graphs, are odd harmonious graphs.