Odd harmonious labeling of two graphs containing star

Abstract

An odd harmonious labeling of a graph G is an injective function f:V(G)→{ 0,1,2,…,2| E(G) |−1 } such that the induced function f*:E(G)→{ 1,3,…,2| E(G) |−1 } defined by f*(xy)=f(x)+f(y) is a bijection. A graph that admits odd harmonious labeling is called an odd harmonious graph. The concept of odd harmonious labeling was initiated by Liang and Bai in 2009. By the result of Liang and Bai, a star is an odd harmonious graph. Motivated by a result, we prove that two graphs containing star are still odd harmonious. In this case, we prove that a double stars is an odd harmonious graph. The remaining we prove that an even cycle and a star which is sharing a common vertex is also an odd harmonious graph.