Z2-indices and Hedetniemi's conjecture

TL;DR

This paper explores the relationship between the $ ext{Z}_2$-index of product spaces and Hedetniemi's conjecture, suggesting that if the conjecture holds, then the index of a product equals the minimum of the indices.

Contribution

It establishes a conditional link between Hedetniemi's conjecture and the behavior of the $ ext{Z}_2$-index under product operations on finite $ ext{Z}_2$-complexes.

Findings

If Hedetniemi's conjecture is true, then $ ext{ind}(X imes Y) = ext{min}( ext{ind}(X), ext{ind}(Y))$ for finite $ ext{Z}_2$-complexes.

The paper connects a graph theory conjecture with topological index properties.

Provides a conditional equivalence linking combinatorial and topological properties.

Abstract

The $\mathbb{Z}_2$-index ${\rm ind}(X)$ of a $\mathbb{Z}_2$-CW-complex $X$ is the smallest number $n$ such that there is a $\mathbb{Z}_2$-map from $X$ to $S^n$. Here we consider $S^n$ as a $\mathbb{Z}_2$-space by the antipodal map. Hedetniemi's conjecture is a long standing conjecture in graph theory concerning the graph coloring problem of tensor products of finite graphs. We show that if Hedetniemi's conjecture is true, then ${\rm ind}(X \times Y) = \min \{ {\rm ind}(X) , {\rm ind}(Y)\}$ for every pair $X$ and $Y$ of finite $\mathbb{Z}_2$-complexes.