VERTEX DISTINGUISHING PROPER EDGE COLORINGS OF THE JOIN GRAPHS - Petrosyan, Tigran K. (Proceedings of the YSU A: Physical and Mathematical Sciences, 2025)
A proper edge coloring of a graph $G$ is a mapping $f:E(G)\longrightarrow \mathbb{Z}_{≥ 0}$ such that $f(e) ot=f(e')$ 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 $u,v \in V(G)$, $S(u,f) e S(v,f)$, 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 $χ'_{vd}(G)$ 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.
Keywords: edge coloring; proper edge coloring; vertex distinguishing; proper coloring; join graphs