Odd harmonious labeling of some new families of graphs

Abstract

A graph $G\left(p,q\right)$ is said to be odd harmonious if there exists an injection $f:V\left(G\right)\to \left\{0,1,2,\cdots ,2q-1\right\}$ such that the induced function $f^{*}:E\left(G\right)\to \left\{1,3,\cdots ,2q-1\right\}$ defined by $f^{*}\left(uv\right)=f\left(u\right)+f\left(v\right)$ is a bijection. A graph that admits odd harmonious labeling is called odd harmonious graph. In this paper, we prove that shadow and splitting of graph $K_{2,n,}{C}_n$ for $n\equiv 0$ (mod 4), the graph $H_{n,n},$ double quadrilateral snakes $DQ\left(n\right),n\ge 2,$ the graph $P_{r,m}$ if $m$ is odd, banana tree and the path union of cycles $C_n$ for $n\equiv 0$ (mod 4) are odd harmonious.