Subchromatic numbers of powers of graphs with excluded minors

TL;DR

The paper addresses bounding the subchromatic number $χ_{sub}(G)$ of graph powers $G^{d}$, with a focus on planar graphs and broader minor-closed classes. It introduces the semi-weak colouring number $swcol_k(G,σ)$ and proves $χ_{sub}(G^{d}) ≤ swcol_{2d}(G,σ)$ (and $≤ swcol_{2d-1}(G,σ)$ for odd $d$), enabling tighter bounds than previously known. Using this framework, the authors achieve $χ_{sub}(G^{2}) ≤ 43$ for planar graphs (with sharper constants for large girth) and $χ_{sub}(G^{3}) ≤ 95$, by extending decompositions of planar graphs and leveraging girth-based refinements; they further derive bounds for $χ_{sub}(G^{p})$ when $G$ has bounded treewidth, bounded simple treewidth, bounded genus, or excludes certain minors. Additionally, they provide a polynomial-time $2$-approximation algorithm for the subchromatic number in classes with bounded layered cliquewidth, enabling practical approximations for powers of planar graphs even when the base graph or the power is not provided.

Abstract

A $k$-subcolouring of a graph $G$ is a function $f:V(G) \to \{0,\ldots,k-1\}$ such that the set of vertices coloured $i$ induce a disjoint union of cliques. The subchromatic number, $χ_{\textrm{sub}}(G)$, is the minimum $k$ such that $G$ admits a $k$-subcolouring. Nešetřil, Ossona de Mendez, Pilipczuk, and Zhu (2020), recently raised the problem of finding tight upper bounds for $χ_{\textrm{sub}}(G^2)$ when $G$ is planar. We show that $χ_{\textrm{sub}}(G^2)\le 43$ when $G$ is planar, improving their bound of 135. We give even better bounds when the planar graph $G$ has larger girth. Moreover, we show that $χ_{\textrm{sub}}(G^{3})\le 95$, improving the previous bound of 364. For these we adapt some recent techniques of Almulhim and Kierstead (2022), while also extending the decompositions of triangulated planar graphs of Van den Heuvel, Ossona de Mendez, Quiroz, Rabinovich and Siebertz (2017), to planar graphs of arbitrary girth. Note that these decompositions are the precursors of the graph product structure theorem of planar graphs. We give improved bounds for $χ_{\textrm{sub}}(G^p)$ for all $p$, whenever $G$ has bounded treewidth, bounded simple treewidth, bounded genus, or excludes a clique or biclique as a minor. For this we introduce a family of parameters which form a gradation between the strong and the weak colouring numbers. We give upper bounds for these parameters for graphs coming from such classes. Finally, we give a 2-approximation algorithm for the subchromatic number of graphs coming from any fixed class with bounded layered cliquewidth. In particular, this implies a 2-approximation algorithm for the subchromatic number of powers $G^p$ of graphs coming from any fixed class with bounded layered treewidth (such as the class of planar graphs). This algorithm works even if the power $p$ and the graph $G$ is unknown.

Theorems & Definitions (9)

• Theorem 1.1: Nešetřil et al. clusteringpowers
• Theorem 1.2
• Theorem 1.3: Van den Heuvel et al. vandenHeuveletal2017
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Lemma 2.2
• proof
• Claim 1