, Volume 5, Issue 2, pp 121–126 | Cite as

The chromatic number of the product of two 4-chromatic graphs is 4

  • Mohamed El-Zahar
  • Norbert Sauer


For any graphG and numbern≧1 two functionsf, g fromV(G) into {1, 2, ...,n} are adjacent if for all edges (a, b) ofG, f(a)g(b). The graph of all such functions is the colouring graph ℒ(G) ofG. We establish first that χ(G)=n+1 implies χ(ℒ(G))=n iff χ(G ×H)=n+1 for all graphsH with χ(H)≧n+1. Then we will prove that indeed for all 4-chromatic graphsG χ(ℒ(G))=3 which establishes Hedetniemi’s [3] conjecture for 4-chromatic graphs.

AMS subject classification (1980)

05 C 15 


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  1. [1]
    S. A. Burr, P. Erdős andL. Lovász, On graphs of Ramsey type,Ars Comb. 1 (1976), 167–190.zbMATHGoogle Scholar
  2. [2]
    D. Duffus, B. Sands andR. E. Woodrow, On the Chromatic Number of the Product of Graphs,Journal of Graph Theory, to appear.Google Scholar
  3. [3]
    S. T. Hedetniemi, Homomorphisms of graphs and automata,Univ. of Michigan Technical Report 03105-44-T, 1966.Google Scholar

Copyright information

© Akadémiai Kiadó 1985

Authors and Affiliations

  • Mohamed El-Zahar
    • 1
  • Norbert Sauer
    • 1
  1. 1.Department of Mathematics and StatisticsUniversity of CalgaryCalgaryCanada

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