Hedetniemi's Conjecture Via Altermatic Number

In 2015, the present authors introduced the altermatic number and the strong altermatic number of graphs and showed that they provide tight lower bounds for the chromatic number. In this paper, we consider Hedetniemi's conjecture (1966) which asserts that the chromatic number of the Categorical product of two graphs is equal to the minimum of their chromatic numbers. A family of graphs is tight if Hedetniemi's conjecture holds for any two graphs of this family. By topological methods, it has earlier been shown that the family of Schrijver graphs and also the iterated Mycielskian of Schrijver graphs are tight. We prove some relaxation versions of Hedetniemi's conjecture for the altermatic number and the strong altermatic number. As a consequence, we enrich the aforementioned tight family to other graphs such as a large family of Kneser multigraphs, matching graphs, permutation graphs, and the iterated Mycielskian of any such graphs. Moreover, by presenting a generalization of Gale's lemma, we show that the altermatic number and the storng altermatic number provide tight lower bounds for some well-known topological parameters attached to graphs via the Borsuk-Ulam theorem.