Acyclic coloring of graphs

  • Authors:

  • Noga Alon;Colin Mcdiarmid;Bruce Reed

  • Affiliations:

  • Department of Mathematics, Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, Israel and IBM Almaden Research Center, San Jose, CA 95120;Department of Statistics, University of Oxford, Oxford, England;Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Canada

  • Venue:
  • Random Structures & Algorithms
  • Year:
  • 1991

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Abstract

A vertex coloring of a graph G is called acyclic if no two adjacent vertices have the same color and there is no two-colored cycle in G. The acyclic chromatic number of G, denoted by A(G), is the least number of colors in an acyclic coloring of G. We show that if G has maximum degree d, then A(G) = 0(d4/3) as d → ∞. This settles a problem of Erdös who conjectured, in 1976, that A(G) = o(d2) as d → ∞. We also show that there are graphs G with maximum degree d for which A(G) = Ω(d4/3/(log d)1/3); and that the edges of any graph with maximum degree d can be colored by 0(d) colors so that no two adjacent edges have the same color and there is no two-colored cycle. All the proofs rely heavily on probabilistic arguments. © 1991 Wiley Periodicals, Inc.