Acyclic coloring of graphs

Noga Alon, Colin Mcdiarmid, Bruce Reed

Research output: Contribution to journalArticlepeer-review

190 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)277-288
Number of pages12
JournalRandom Structures & Algorithms
Issue number3
StatePublished - 1991
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Software
  • General Mathematics
  • Computer Graphics and Computer-Aided Design
  • Applied Mathematics


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