On Vertex-Distinguishing Proper Edge Colorings of Graphs Satisfying the Ore Condition

Abstract

An edge coloring is proper if no two adjacent edges are assigned the same color and vertex-distinguishing proper coloring if it is proper and incident edge sets of every two distinct vertices are assigned different sets of colors. The minimum number of colors required for a vertex-distinguishing proper edge coloring of a simple graph $G$ is denoted by ${\overline{\chi }}^{\prime }(G)$. In this paper, we prove that ${\overline{\chi }}^{\prime }(G)\le \mathrm{\Delta }(G)+4$ if $G=(V,E)$ is a connected graph of order $n\ge 3$ and ${\sigma }_2(G)\ge n$, where ${\sigma }_2(G)=min\{d(x)+d(y)|xy\in E(G)\}$.