A Characterization on the Adjacent Vertex Distinguishing Index of Planar Graphs with Large Maximum Degree

Abstract

An adjacent vertex distinguishing coloring of a graph $G$ is a proper edge coloring of $G$ such that any pair of adjacent vertices admits different sets of colors. The minimum number of colors needed for such a coloring of $G$ is denoted by $\chi'_a(G)$. In this paper, we show that if $G$ is a planar graph with maximum degree $\Delta\ge 16$, then $\Delta\le \chi'_{a}(G)\le \Delta+1$, and $\chi'_a(G)=\Delta+1$ if and only if $G$ contains two adjacent vertices of maximum degree.