# Coloring Properties of Categorical Product of General Kneser Hypergraphs

## TL;DR

This paper investigates the coloring properties of the categorical product of general Kneser hypergraphs, providing new bounds and extending results related to Hedetniemi's conjecture and Zhu's generalization.

## Contribution

It introduces new colorful coloring results and a significantly improved lower bound for the chromatic number of these hypergraphs, advancing understanding of Zhu's conjecture.

## Key findings

- New colorful coloring results for Kneser hypergraph products
- A novel lower bound surpassing previous bounds
- Extended classes of hypergraphs satisfying Zhu's conjecture

## Abstract

More than 50 years ago Hedetniemi conjectured that the chromatic number of categorical product of two graphs is equal to the minimum of their chromatic numbers. This conjecture has received a considerable attention in recent years. Hedetniemi's conjecture were generalized to hypergraphs by Zhu in 1992. Hajiabolhassan and Meunier (2016) introduced the first nontrivial lower bound for the chromatic number of categorical product of general Kneser hypergraphs and using this lower bound, they verified Zhu's conjecture for some families of hypergraphs. In this paper, we shall present some colorful type results for the coloring of categorical product of general Kneser hypergraphs, which generalize the Hajiabolhassan-Meunier result. Also, we present a new lower bound for the chromatic number of categorical product of general Kneser hypergraphs which can be extremely better than the Hajiabolhassan-Meunier lower bound. Using this lower bound, we enrich the family of hypergraphs satisfying Zhu's conjecture.