Acyclic edge colorings of graphs

  • Authors:

  • Noga Alon;Benny Sudakov;Ayal Zaks

  • Affiliations:

  • Department of Mathematics, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University Tel Aviv, Israel;Department of Mathematics, Princeton University, and Institute for Advanced Study, Princeton, New Jersey 08540;Department of Statistics and Operations Research, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, Israel

  • Venue:
  • Journal of Graph Theory
  • Year:
  • 2001

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Abstract

A proper coloring of the edges of a graph G is called acyclic if there is no 2-colored cycle in G. The acyclic edge chromatic number of G, denoted by a′(G), is the least number of colors in an acyclic edge coloring of G. For certain graphs G, a′(G) ≥ Δ(G) + 2 where Δ(G) is the maximum degree in G. It is known that a′(G) ≤ 16 Δ(G) for any graph G. We prove that there exists a constant c such that a′(G) ≤ Δ(G) + 2 for any graph G whose girth is at least cΔ(G) log Δ(G), and conjecture that this upper bound for a′(G) holds for all graphs G. We also show that a′(G) ≤ Δ + 2 for almost all Δ-regular graphs. © 2001 John Wiley & Sons, Inc. J Graph Theory 37: 157–167, 2001