On Hedetniemi's conjecture and the colour template scheme (Q1613520)

Hedetniemi's conjecture states that the chromatic number of a product of graphs is the minimum of the chromatic numbers of the factors. It is shown that if \(G\) and \(H\) are connected graphs of chromatic number at least 5 and both contain odd wheels as subgraphs, then their product also has chromatic number at least 5. The paper also provides counterexamples to the conjecture of El-Zahar and Saucer on the structure of \(n\)-chromatic subgraphs of products of \(n\)-chromatic graphs, implying that their method is not sufficient to prove all cases of Hedetniemi's conjecture for chromatic number larger than 4.