Adjacent Vertex Distinguishing Edge‐Colorings

Abstract

An adjacent vertex distinguishing edge-coloring of a simple graph $G$ is a proper edge-coloring of $G$ such that no pair of adjacent vertices meets the same set of colors. The minimum number of colors $\chi^\prime_a(G)$ required to give $G$ an adjacent vertex distinguishing coloring is studied for graphs with no isolated edge. We prove $\chi^\prime_a(G)\le5$ for such graphs with maximum degree $\Delta(G)=3$ and prove $\chi^\prime_a(G)\le\Delta(G)+2$ for bipartite graphs. These bounds are tight. For $k$-chromatic graphs $G$ without isolated edges we prove a weaker result of the form $\chi^\prime_a(G)=\Delta(G)+O(\log k)$.