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


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.

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    S. T. Hedetniemi, Homomorphisms of graphs and automata,Univ. of Michigan Technical Report 03105-44-T, 1966.

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This research was supported by NSERC grant A7213

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El-Zahar, M., Sauer, N. The chromatic number of the product of two 4-chromatic graphs is 4. Combinatorica 5, 121–126 (1985).

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AMS subject classification (1980)

  • 05 C 15