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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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
AMS subject classification (1980)
- 05 C 15
- 04 A 05
- 04 A 10
- 04 A 20