Hedetniemi's conjecture from the topological viewpoint

TL;DR

This paper investigates a topological approach to Hedetniemi's conjecture, confirming it for certain cases and providing new bounds for graph coloring problems using topological indices.

Contribution

The paper establishes a stronger formula for the homological index of joins of $ ext{Z}/2$-spaces and confirms Hedetniemi's conjecture in specific cases, including when one factor is an n-sphere.

Findings

Confirmed the conjecture for the homological index case.

Verified the conjecture when one factor is an n-sphere.

Provided new topological bounds for graph chromatic numbers.

Abstract

This paper is devoted to studying a topological version of the famous Hedetniemi conjecture which says: The $\mathbb Z/2$-index of the Cartesian product of two $\mathbb Z/2$-spaces is equal to the minimum of their $\mathbb Z/2$-indexes. We fully confirm the version of this conjecture for the homological index via establishing a stronger formula for the homological index of the join of $\mathbb Z/2$-spaces. Moreover, we confirm the original conjecture for the case when one of the factors is an $n$-sphere. Analogous results for $\mathbb Z/p$-spaces are presented as well. In addition, we answer a question about computing the index of some non-trivial products, raised by Marcin Wrochna. Finally, some new topological lower bounds for the chromatic number of the Categorical product of (hyper-)graphs are presented.