On the Adjacent Vertex Distinguishing Edge Chromatic Number of Graphs

Abstract

An adjacent vertex distinguishing edge coloring, or an avd-coloring, of a simple graph $G$ is a proper edge coloring of $G$ such that for any two adjacent and distinct vertices $u$ and $v$ in $G$, the set of colors assigned to the edges incident to $u$ differs from the set of colors assigned to the edges incident to $v$. In this paper, we prove that graphs with maximum degree $3$ and with no isolated edges partly satisfy the adjacent vertex distinguishing edge coloring conjecture.