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 $\chi^\prime_a(G)=\Delta(G)+O(\log k)$.