Upper bounds for adjacent vertex-distinguishing edge coloring

Abstract

An adjacent vertex-distinguishing edge coloring of a graph is a proper edge coloring such that no pair of adjacent vertices meets the same set of colors. The adjacent vertex-distinguishing edge chromatic number is the minimum number of colors required for an adjacent vertex-distinguishing edge coloring, denoted as $\chi '_{as}(G)$. In this paper, we prove that for a connected graph G with maximum degree $\Delta \ge 3$, $\chi '_{as}(G)\le 3\Delta -1$, which proves the previous upper bound. We also prove that for a graph G with maximum degree $\Delta \ge 458$ and minimum degree $\delta \ge 8\sqrt{\Delta ln \Delta }$, $\chi '_{as}(G)\le \Delta +1+5\sqrt{\Delta ln \Delta }$.