Weak diameter coloring of graphs on surfaces

TL;DR

This paper proves that graphs on surfaces can be colored from lists to ensure monochromatic components have bounded weak diameter, extending known results and solving a problem for planar triangle-free graphs.

Contribution

It introduces a list-coloring approach for graphs on surfaces that guarantees bounded weak diameter of monochromatic components, addressing open problems.

Findings

Monochromatic subgraphs have bounded weak diameter in the list-coloring setting.

Results apply to graphs on fixed surfaces, including planar triangle-free graphs.

Extends previous results to the list-coloring context, answering open questions.

Abstract

Consider a graph $G$ drawn on a fixed surface, and assign to each vertex a list of colors of size at least two if $G$ is triangle-free and at least three otherwise. We prove that we can give each vertex a color from its list so that each monochromatic connected subgraph has bounded weak diameter (i.e., diameter measured in the metric of the whole graph $G$, not just the subgraph). In case that $G$ has bounded maximum degree, this implies that each connected monochromatic subgraph has bounded size. This solves a problem of Esperet and Joret for planar triangle-free graphs, and extends known results in the general case to the list setting, answering a question of Wood.