Centered colorings in minor-closed graph classes

TL;DR

This paper proves that for any fixed minor-closed graph class excluding a complete graph, there exists a bounded-coloring scheme called p-centered coloring, with the number of colors depending polynomially on p and the size of the excluded minor.

Contribution

It establishes a polynomial bound on the number of colors needed for p-centered colorings in K_t-minor-free graphs, extending understanding of colorings in minor-closed classes.

Findings

K_t-minor-free graphs admit p-centered colorings with O(p^{t-1}) colors

The result generalizes previous bounds for specific minor-closed classes

Provides a new tool for graph coloring in minor-closed graph classes

Abstract

A vertex coloring $\varphi$ of a graph $G$ is $p$-centered if for every connected subgraph $H$ of $G$, either $\varphi$ uses more than $p$ colors on $H$, or there is a color that appears exactly once on $H$. We prove that for every fixed positive integer $t$, every $K_t$-minor-free graph admits a $p$-centered coloring using $\mathcal{O}(p^{t-1})$ colors.