Clustered colouring of graph classes with bounded treedepth or pathwidth

TL;DR

This paper investigates the minimum number of colours needed to colour graphs with bounded treedepth or pathwidth so that each colour class has small connected components, providing exact bounds for minor-closed classes.

Contribution

It establishes the exact clustered chromatic number for minor-closed classes with bounded treedepth and provides optimal bounds for classes with bounded pathwidth, also determining fractional values.

Findings

Exact clustered chromatic number for minor-closed classes with bounded treedepth.

Optimal upper bounds for classes with bounded pathwidth.

Determination of fractional clustered chromatic number for all minor-closed classes.

Abstract

The "clustered chromatic number" of a class of graphs is the minimum integer $k$ such that for some integer $c$ every graph in the class is $k$-colourable with monochromatic components of size at most $c$. We determine the clustered chromatic number of any minor-closed class with bounded treedepth, and prove a best possible upper bound on the clustered chromatic number of any minor-closed class with bounded pathwidth. As a consequence, we determine the fractional clustered chromatic number of every minor-closed class.