The chromatic number of the product of two ℵ1-chromatic graphs can be countable

Abstract

We prove (in ZFC) that for every infinite cardinal ϰ there are two graphsG 0,G 1 with χ(G 0)=χ(G 1)=ϰ+ and χ(G 0×G 1)=ϰ. We also prove a result from the other direction. If χ(G 0)≧≧ℵ0 and χ(G 1)=k<ω, then χ(G 0×G 1)=k.

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References

  1. [1]

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

  2. [2]

    S. T. Hedetniemi, Homomorphisms of graphs and automata,Univ. of Michigan Technical Report 03 105-44-T, 1966.

  3. [3]

    S. Todorčević, Stationary sets, Trees and Continuums,Publ. Inst. Math. (Beograd) 27 (41) (1981), 249–262.

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Hajnal, A. The chromatic number of the product of two ℵ1-chromatic graphs can be countable. Combinatorica 5, 137–139 (1985). https://doi.org/10.1007/BF02579376

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AMS subject classification (1980)

  • 05 C 15
  • 04 A 05
  • 04 A 10
  • 04 A 20