On Hedetniemi's conjecture and the colour template scheme

Abstract

Hedetniemi's conjecture states that the chromatic number of a product of graphs is the minimum of the chromatic numbers of the factors. El-Zahar and Sauer (Combinatorica 5 (1985) 121) have shown that the chromatic number of a product of 4-chromatic graphs is 4. In this paper, we investigate to which extent their methods can be adapted to the case of 5-chromatic graphs. Our main result is that if G and H are connected graphs of chromatic number at least 5 that both contain odd wheels as subgraphs, then their product also has chromatic number at least 5. However, we also provide counterexamples to the El-Zahar and Sauer conjecture on the structure of n-chromatic subgraphs of products of n-chromatic graphs. This implies that the method of El-Zahar and Sauer will not be sufficient to prove all cases of Hedetniemi's conjecture for chromatic numbers larger than 4.