Hedetniemi’s Conjecture Via Alternating Chromatic Number

In an earlier paper, the present authors (2013) [1] introduced the alternating chromatic number for hypergraphs and used Tucker’s Lemma, an equivalent combinatorial version of the Borsuk-Ulam Theorem, to show that the alternating chromatic number is a lower bound for the chromatic number. In this paper, we determine the chromatic number of some families of graphs by specifying their alternating chromatic number. We define matching-dense graph as a graph whose vertex set is the set of all matchings of a specified size of a dense graph and two vertices are adjacent if the corresponding matchings are edge-disjoint. We determine the chromatic number of matching-dense graphs in terms of the generalized Turán number of matchings. Also, we consider Hedetniemi’s conjecture which asserts that the chromatic number of the Categorical product of two graphs is equal to the minimum of their chromatic numbers. By topological methods, it has earlier been shown that Hedetniemi’s conjecture holds for any two graphs of the family of Kneser graphs, Schrijver graphs, and the iterated Mycielskian of any such graphs. We extend these results to other graphs such as a large family of Kneser multigraphs, matching graphs, and permutation graphs.