Cooperative colorings of forests

TL;DR

This paper investigates the limits of cooperative colorings in forests, showing that for large maximum degree, certain families of star forests cannot be cooperatively colored, improving previous bounds significantly.

Contribution

It establishes new asymptotic bounds on the size of forest families that admit no cooperative coloring, especially for star forests, advancing understanding in graph coloring theory.

Findings

Existence of large forest families with no cooperative coloring for high degree

Improved bounds on the number of forests needed for cooperative coloring

Optimality of the bounds for star forests

Abstract

Given a family $\mathcal G$ of graphs spanning a common vertex $V$, a cooperative coloring of $\mathcal G$ is a collection of one independent set from each graph of $\mathcal G$ such that the union of these independent sets equals $V$. We prove that when $d$ is large, there exists a family $\mathcal G$ of $(1+o(1)) \frac{\log d}{\log \log d}$ forests of maximum degree $d$ that admits no cooperative coloring, which significantly improves a result of Aharoni, Berger, Chudnovsky, Havet, and Jiang (Electronic Journal of Combinatorics, 2020). Our family $\mathcal G$ consists entirely of star forests, and we show that this value for $|\mathcal G|$ is asymptotically best possible in the case that $\mathcal G$ is a family of star forests.