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.