Defective and Clustered Colouring of Graphs with Given Girth

TL;DR

This paper investigates the defective and clustered chromatic numbers of graphs with specified girth within various minor-closed classes, revealing that for certain classes, these numbers coincide and depend on parameters like treewidth and treedepth.

Contribution

It establishes bounds and exact values for defective and clustered chromatic numbers in graphs with given girth across multiple minor-closed classes, including new equalities for triangle-free graphs with bounded treewidth and treedepth.

Findings

Defective, clustered, and chromatic numbers are equal for triangle-free graphs with bounded treewidth.

Similar equality holds for graphs with bounded treedepth.

Results generalize to graphs excluding complete subgraphs $K_p$.

Abstract

The defective chromatic number of a graph class $\mathcal{G}$ is the minimum integer $k$ such that for some integer $d$, every graph in $\mathcal{G}$ is $k$-colourable such that each monochromatic component has maximum degree at most $d$. Similarly, the clustered chromatic number of a graph class $\mathcal{G}$ is the minimum integer $k$ such that for some integer $c$, every graph in $\mathcal{G}$ is $k$-colourable such that each monochromatic component has at most $c$ vertices. This paper determines or establishes bounds on the defective and clustered chromatic numbers of graphs with given girth in minor-closed classes defined by the following parameters: Hadwiger number, treewidth, pathwidth, treedepth, circumference, and feedback vertex number. One striking result is that for any integer $k$, for the class of triangle-free graphs with treewidth $k$, the defective chromatic number, clustered chromatic number and chromatic number are all equal. The same result holds for graphs with treedepth $k$, and generalises for graphs with no $K_p$ subgraph.