Hedetniemi's conjecture for Kneser hypergraphs

TL;DR

This paper proves Zhu's hypergraph extension of Hedetniemi's conjecture for Kneser hypergraphs of the same rank, establishing new cases where the conjecture holds and providing a lower bound on their chromatic number.

Contribution

It confirms Zhu's conjecture for Kneser hypergraphs of the same rank and introduces a general lower bound on their chromatic number using the $Z_p$-Tucker lemma.

Findings

Zhu's conjecture holds for Kneser hypergraphs of the same rank.

A new lower bound on the chromatic number of the categorical product is established.

Identifies new families of graphs satisfying Hedetniemi's conjecture.

Abstract

One of the most famous conjecture in graph theory is Hedetniemi's conjecture stating that the chromatic number of the categorical product of graphs is the minimum of their chromatic numbers. Using a suitable extension of the definition of the categorical product, Zhu proposed in 1992 a similar conjecture for hypergraphs. We prove that Zhu's conjecture is true for the usual Kneser hypergraphs of same rank. It provides to the best of our knowledge the first non-trivial and explicit family of hypergraphs with rank larger than two satisfying this conjecture (the rank two case being Hedetniemi's conjecture). We actually prove a more general result providing a lower bound on the chromatic number of the categorical product of any Kneser hypergraphs as soon as they all have same rank. We derive from it new families of graphs satisfying Hedetniemi's conjecture. The proof of the lower bound relies on the $Z_p$-Tucker lemma.