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 \(\overline{\chi}'(G)\). In this paper, we prove that \(\overline{\chi}'(G) \leq \Delta(G) + {4}\) if \(G = (V, E)\) is a connected graph of order \(n \geq 3\) and \(\sigma_2(G) \geq n\), where \(\sigma_2(G) = \min\{d(x) + d(y) | xy \in E(G)\}\).