Adjacent vertex-distinguishing edge coloring of 2-degenerate graphs

Abstract

The adjacent vertex-distinguishing chromatic index $\chi '_{avd}(G)$ of a graph $G$ is the smallest integer $k$ for which $G$ admits a proper edge $k$-coloring such that no pair of adjacent vertices are incident with the same set of colors. In this paper, we prove that if $G$ is a $2$-degenerate graph without isolated edges, then $\chi '_{avd} (G)\le \max \{6, \Delta (G)+1\}$. Moreover, we also show that when $\Delta \ge 6$, $\chi '_{avd}= \Delta (G)+1$ if and only if $G$ contains two adjacent vertices of maximum degree.