Hedetniemi's conjecture from the topological viewpoint

Abstract

This paper is devoted to studying some topological questions related to well known Hedetniemi's conjecture on the chromatic number of the categorical product of graphs. It is surprisingly shown that Hedetniemi's conjecture implies a statement in Topology by M. Wrochna and T. Matsushita (independently). Precisely, they showed if Hedetniemi's conjecture is true, then the Z 2 -index of the Cartesian product of two Z 2 -spaces is equal to the minimum of their Z 2 -indexes. In addition, Wrochna conjectured the correctness of the latter statement. We show Wrochna's conjecture is true for the case that one of the Z 2 -spaces is an n-sphere S n , and fully confirm the version of this conjecture for the homological index.There is also a generalization of Hedetniemi's conjecture for hypergraphs. To approach this conjecture, we establish a topological lower bound for the chromatic number of the categorical product of hypergraphs. Consequently, we enrich the family of known hypergraphs satisfying this conjecture. Especially, we reprove the known fact that the conjecture is valid for the usual Kneser r-hypergraphs. We also give some other applications of this bound, such as, a new proof of a well-known conjecture of Erdős, and a new way for constructing r-hypergraphs with arbitrary high chromatic numbers and small clique numbers.