Cooperative colorings of trees and of bipartite graphs

TL;DR

This paper investigates the minimal number of graphs needed to guarantee a cooperative coloring in systems of trees and bipartite graphs with bounded maximum degree, establishing bounds that grow logarithmically with degree.

Contribution

It provides new bounds on the number of graphs required for cooperative colorings in trees and bipartite graphs, advancing understanding of their combinatorial properties.

Findings

For trees, the minimal number grows between logarithmic and double logarithmic in degree.

For bipartite graphs, the bounds are between logarithmic and linear in degree.

The results establish tight bounds up to constant factors for these classes.

Abstract

Given a system $(G_1, \ldots ,G_m)$ of graphs on the same vertex set $V$, a cooperative coloring is a choice of vertex sets $I_1, \ldots ,I_m$, such that $I_j$ is independent in $G_j$ and $\bigcup_{j=1}^{m}I_j = V$. For a class $\mathcal{G}$ of graphs, let $m_{\mathcal{G}}(d)$ be the minimal $m$ such that every $m$ graphs from $\mathcal{G}$ with maximum degree $d$ have a cooperative coloring. We prove that $\Omega(\log\log d) \le m_\mathcal{T}(d) \le O(\log d)$ and $\Omega(\log d)\le m_\mathcal{B}(d) \le O(d/\log d)$, where $\mathcal{T}$ is the class of trees and $\mathcal{B}$ is the class of bipartite graphs.