Immersion and clustered coloring

TL;DR

This paper determines the minimum number of colors needed for clustered coloring of graphs excluding a fixed graph $H$ as an immersion, extending classical conjectures and introducing a new reduction technique via tree decompositions.

Contribution

It establishes tight bounds on the number of colors for $H$-immersion free graphs and develops a novel lemma linking clustering coloring to tree decompositions.

Findings

Exact bounds for $H$-immersion free graphs' coloring

A new lemma reduces clustering coloring to tree decomposition problems

Unified proof of existing results on clustered coloring in minor-closed families

Abstract

Hadwiger and Haj\'{o}s conjectured that for every positive integer $t$, $K_{t+1}$-minor free graphs and $K_{t+1}$-topological minor free graphs are properly $t$-colorable, respectively. Clustered coloring version of these two conjectures which only require monochromatic components to have bounded size has been extensively studied. In this paper we consider the clustered coloring version of the immersion-variant of Hadwiger's and Haj\'{o}s' conjecture proposed by Lescure and Meyniel and independently by Abu-Khzam and Langston. We determine the minimum number of required colors for $H$-immersion free graphs, for any fixed graph $H$, up to a small additive absolute constant. Our result is tight for infinitely many graphs $H$. A key machinery developed in this paper is a lemma that reduces a clustering coloring problem on graphs to the one on the torsos of their tree-cut decomposition or tree-decomposition. A byproduct of this machinery is a unified proof of a result of Alon, Ding, Oporowski and Vertigan and a result of the author and Oum about clustered coloring graphs of bounded maximum degree in minor-closed families.