Acyclic edge coloring of graphs

Abstract

An acyclic edge coloring of a graph G is a proper edge coloring such that the subgraph induced by any two color classes is a linear forest (an acyclic graph with maximum degree at most two). The acyclic chromatic indexχa′(G) of a graph G is the least number of colors needed in an acyclic edge coloring of G. Fiamčík (1978) conjectured that χa′(G)≤Δ(G)+2, where Δ(G) is the maximum degree of G. This conjecture is well known as the Acyclic Edge Coloring Conjecture (AECC). A graph G with maximum degree at most κ is κ-deletion-minimal if χa′(G)>κ and χa′(H)≤κ for every proper subgraph H of G. The purpose of this paper is to provide many structural lemmas on κ-deletion-minimal graphs. By using the structural lemmas, we firstly prove that AECC is true for the graphs with maximum average degree less than four (Theorem 4.3). We secondly prove that AECC is true for the planar graphs without triangles adjacent to cycles of length at most four, with an additional condition that every 5-cycle has at most three edges contained in triangles (Theorem 4.4), from which we can conclude some known results as corollaries. We thirdly prove that every planar graph G without intersecting triangles satisfies χa′(G)≤Δ(G)+3 (Theorem 4.6). Finally, we consider one extreme case and prove it: if G is a graph with Δ(G)≥3 and all the 3+-vertices are independent, then χa′(G)=Δ(G). We hope the structural lemmas will shed some light on the acyclic edge coloring problems.