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The chromatic number of the product of two 4-chromatic graphs is 4. (English) Zbl 0575.05028
The product $$G\times H$$ of two graphs G and H is the graph whose vertex set is $$V(G)\times V(H)$$ and whose edge set consists of all pairs $$((a,b), (\bar a,\bar b))$$ such that $$(a,\bar a)$$ and $$(b,\bar b)$$ are edges of G and H respectively. The chromatic number of the product is early seen to satisfy the bound $$\chi(G\times H)\leq \min\{\chi(G),\chi(H)\}$$. S. Hedetniemi [Homomorphisms of graphs and automata, Univ. of Michigan Technical report 03105-44-T (1966)] has conjectured that equality holds in this bound, verifying this when the right-hand side is 3. The present paper verifies this conjecture when the right-hand side is 4.
Reviewer: J.G.Oxley

##### MSC:
 05C15 Coloring of graphs and hypergraphs
##### Keywords:
product of graphs; chromatic number
Full Text:
##### References:
 [1] S. A. Burr, P. Erdos andL. Lovász, On graphs of Ramsey type,Ars Comb. 1 (1976), 167–190. [2] D. Duffus, B. Sands andR. E. Woodrow, On the Chromatic Number of the Product of Graphs,Journal of Graph Theory, to appear. · Zbl 0664.05019 [3] S. T. Hedetniemi, Homomorphisms of graphs and automata,Univ. of Michigan Technical Report 03105-44-T, 1966.
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