Odd harmonious labeling on squid graph and double squid graph

Abstract

An injective function f from set of vertices in graph G to a set of {0,1,…,|E| − 1} is called an odd harmonious labeling if the function f induced the edge function f* from the set of edges of G to a set of odd positive integer number {1,3,5,…,2|E| − 1} with f*(xy) = f(x) + f(y) for every edge xy in E. Graph that has an odd harmonious labeling is called odd harmonious graph. The squid graph Tn,k is a graph which is obtained from a cycle Cn and we add k pendant to one vertex of the cycle. It is known that Cn is an odd harmonious graph if and only if n = 0 mod 4. However, by adding at least one pendant in the cycle graph, we can label the new graph odd harmoniously for all even number of vertices. In this paper, we showed that the graph Tn,k and T 2n,k are an odd harmonious graph, for n = 0 (mod 2), n ≥ 4 and k ≥ 1. The construction of the odd harmonious labeling of the graph Tn,k and T 2n,k are inspired by the odd harmonious labeling of Cn for n = 0(mod 4).