Odd harmonious labeling of some family of snake graphs

Graph labeling is a way of assigning integers to vertices or edges of a graph that satisfy certain conditions. One of graph labeling is odd harmonious labeling. Let G = G(p, q) be a graph that have p vertices and q edges. An odd harmonious labeling of G 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*: E(G) → {1, 3, 5, … , 2q - 1}, and f* (uv) = f(u) + f(v) for every edge uv ∈ E(G), is bijective. A snake graph k(G) is a graph obtained from a path on k edges by replacing each edge by a graph isomorphic to G. If such labeling exists, then G is said to be odd harmonious. In this paper we show that snake graph k(G) is odd harmonious for some graph G.