VERTEX DISTINGUISHING PROPER EDGE COLORINGS OF THE JOIN GRAPHS

Abstract

A proper edge coloring of a graph $G$ is a mapping $f:E(G)\longrightarrow \mathbb{Z}_{\geq 0}$ such that $f(e)\not=f(e')$ for every pair of adjacent edges $e$ and $e'$ in $G$. A proper edge coloring $f$ of a graph $G$ is called vertex distinguishing if for any different vertices $u,v \in V(G)$, $S(u,f) \ne S(v,f)$, where $S(v,f) = \{f(e) \ | \ e = wv \in E(G)\}$. The minimum number of colors required for a vertex distinguishing proper coloring of a graph $G$ is denoted by $\chi'_{vd}(G)$ and called vertex distinguishing chromatic index of $G$. In this paper we provide lower and upper bounds on the vertex distinguishing chromatic index of the join graphs.