Cooperative colorings of hypergraphs

TL;DR

This paper introduces the concept of cooperative colorings in hypergraphs, determining their minimal number for specific classes like cycles and paths, and establishing bounds for k-partite hypergraphs with large degrees.

Contribution

It provides exact cooperative chromatic numbers for certain hypergraph classes and bounds for k-partite hypergraphs with high degree, using a new set system partition theorem.

Findings

Exact cooperative chromatic number of 2 for specific hypergraph classes.

Bounds on cooperative chromatic number for k-partite hypergraphs with large degree.

Introduction of a new set system partition theorem.

Abstract

Given a class $\mathcal{H}$ of $m$ hypergraphs ${H}_1, {H}_2, \ldots, {H}_m$ with the same vertex set $V$, a cooperative coloring of them is a partition $\{I_1, I_2, \ldots, I_m\}$ of $V$ in such a way that each $I_i$ is an independent set in ${H}_i$ for $1\leq i\leq m$. The cooperative chromatic number of a class $\mathcal{H}$ is the smallest number of hypergraphs from $\mathcal{H}$ that always possess a cooperative coloring. For the classes of $k$-uniform tight cycles, $k$-uniform loose cycles, $k$-uniform tight paths, and $k$-uniform loose paths, we find that their cooperative chromatic numbers are all exactly two utilizing a new proved set system partition theorem, which also has its independent interests and offers a broader perspective. For the class of $k$-partite $k$-uniform hypergraphs with sufficient large maximum degree $d$, we prove that its cooperative chromatic number has lower bound $\Omega(\log_k d)$ and upper bound $\text{O}\left(\frac{d}{\ln d}\right)^{\frac{1}{k-1}}$.