Islands in minor-closed classes. I. Bounded treewidth and separators

TL;DR

This paper proves that graphs embeddable on any surface have a clustered chromatic number of four, characterizes minor-closed classes with bounded treewidth regarding clustered list chromatic number, and addresses extremal problems in bootstrap percolation.

Contribution

It confirms the conjecture that surface-embeddable graphs have a clustered chromatic number of four and characterizes minor-closed classes with bounded treewidth for clustered list coloring.

Findings

Clustered chromatic number of surface-embeddable graphs is four.

Characterization of minor-closed classes with bounded treewidth for clustered list chromatic number.

Results on extremal problems in bootstrap percolation for minor-closed classes.

Abstract

The clustered chromatic number of a graph class is the minimum integer $t$ such that for some $C$ the vertices of every graph in the class can be colored in $t$ colors so that every monochromatic component has size at most $C$. We show that the clustered chromatic number of the class of graphs embeddable on a given surface is four, proving the conjecture of Esperet and Ochem. Additionally, we study the list version of the concept and characterize the minor-closed classes of graphs of bounded treewidth with given clustered list chromatic number. We further strengthen the above results to solve some extremal problems on bootstrap percolation of minor-closed classes.