Defective and Clustered Colouring of Graphs with Given Girth
Marcin Briański, Robert Hickingbotham, David R. Wood
TL;DR
This work investigates defective and clustered colourings for graphs with given girth within minor‑closed classes, tying these notions to structural graph parameters such as treewidth, treedepth, circumference, and Hadwiger number. It delivers near‑tight bounds and exact equalities in several settings, notably showing that triangle‑free graphs with treewidth $k$ (and with treedepth $k$) have defective, clustered, and proper colourings all equal to $\lceil\frac{k+3}{2}\rceil$, and extending analogous results to $K_p$‑free graphs. The paper develops a robust framework using generalized standard examples, strong colouring numbers, and apex/minor considerations, establishing 2‑colourability with bounded clustering under girth $5$ for broad minor‑closed families, while identifying regimes where the colours must grow with the structural parameter. It also shows Hadwiger‑bounded families admit clustered chromatic numbers equal to the minor parameter and provides improved bounds when girth grows relative to the minor bound, illustrating a deep interaction between girth constraints and minor‑closed structure with implications for Hadwiger’s conjecture variants and related colouring problems.
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.