Clustered Coloring of Graphs with Bounded Layered Treewidth and Bounded Degree

TL;DR

This paper proves that graphs with bounded layered treewidth and maximum degree can be colored with three colors such that each monochromatic component is of polynomial size, improving previous bounds for certain graph classes.

Contribution

It establishes the first polynomial bound on clustering for graphs with layered treewidth and bounded degree, extending results to broader graph classes.

Findings

Graphs with layered treewidth at most k and degree at most Δ are 3-colorable with clustering O(k^{19}Δ^{37}).

This is the first polynomial clustering bound for these graph classes.

The result improves upon previous bounds for graphs of bounded genus.

Abstract

The clustering of a graph coloring is the maximum size of monochromatic components. This paper studies colorings with bounded clustering in graph classes with bounded layered treewidth, which include planar graphs, graphs of bounded Euler genus, graphs embeddable on a fixed surface with a bounded number of crossings per edge, map graphs, amongst other examples. Our main theorem says that every graph with layered treewidth at most $k$ and with maximum degree at most $\Delta$ is $3$-colorable with clustering $O(k^{19}\Delta^{37})$. This is the first known polynomial bound on the clustering. This greatly improves upon a corresponding result of Esperet and Joret for graphs of bounded genus.