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Keywords:

  • acyclic edge coloring;
  • acyclic edge chromatic number;
  • 2-degenerate graphs;
  • series–parallel graphs;
  • outer planar graphs

Abstract

An acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The acyclic chromatic index of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and is denoted by a′(G). A graph is called 2-degenerate if any of its induced subgraph has a vertex of degree at most 2. The class of 2-degenerate graphs properly contains seriesparallel graphs, outerplanar graphs, nonregular subcubic graphs, planar graphs of girth at least 6 and circle graphs of girth at least 5 as subclasses. It was conjectured by Alon, Sudakov and Zaks (and much earlier by Fiamcik) that a′(G)⩽Δ + 2, where Δ = Δ(G) denotes the maximum degree of the graph. We prove the conjecture for 2-degenerate graphs. In fact we prove a stronger bound: we prove that if G is a 2-degenerate graph with maximum degree Δ, then a′(G)⩽Δ + 1. © 2010 Wiley Periodicals, Inc. J Graph Theory 69: 1–27, 2012