On topological relaxations of chromatic conjectures

TL;DR

This paper explores topological relaxations of famous chromatic number conjectures, proving some hold for large graphs and discussing the complexity of the odd Hadwiger conjecture.

Contribution

It introduces topological relaxations for several chromatic conjectures and proves their validity for specific classes of graphs, advancing understanding of these longstanding problems.

Findings

Relaxed topological versions hold for the Behzad-Vizing conjecture.

The odd Hadwiger conjecture is true for large Kneser and Schrijver graphs.

Some conjectures remain difficult in the topological relaxation context.

Abstract

There are several famous unsolved conjectures about the chromatic number that were relaxed and already proven to hold for the fractional chromatic number. We discuss similar relaxations for the topological lower bound(s) of the chromatic number. In particular, we prove that such a relaxed version is true for the Behzad-Vizing conjecture and also discuss the conjectures of Hedetniemi and of Hadwiger from this point of view. For the latter, a similar statement was already proven in an earlier paper of the first author with G. Tardos, our main concern here is that the so-called odd Hadwiger conjecture looks much more difficult in this respect. We prove that the statement of the odd Hadwiger conjecture holds for large enough Kneser graphs and Schrijver graphs of any fixed chromatic number.