Defective and Clustered Choosability of Sparse Graphs

Abstract

An (improper) graph colouring has "defect" $d$ if each monochromatic subgraph has maximum degree at most $d$, and has "clustering" $c$ if each monochromatic component has at most $c$ vertices. This paper studies defective and clustered list-colourings for graphs with given maximum average degree. We prove that every graph with maximum average degree less than $\frac{2d+2}{d+2}k$ is $k$-choosable with defect $d$. This improves upon a similar result by Havet and Sereni [J. Graph Theory, 2006]. For clustered choosability of graphs with maximum average degree $m$, no $(1-\epsilon )m$ bound on the number of colours was previously known. The above result with $d=1$ solves this problem. It implies that every graph with maximum average degree $m$ is $\lfloor \frac{3}{4}m+1\rfloor$-choosable with clustering 2. This extends a result of Kopreski and Yu [Discrete Math., 2017] to the setting of choosability. We then prove two results about clustered choosability that explore the trade-off between the number of colours and the clustering. In particular, we prove that every graph with maximum average degree $m$ is $\lfloor \frac{7}{10}m+1\rfloor$-choosable with clustering $9$, and is $\lfloor \frac{2}{3}m+1\rfloor$-choosable with clustering $O(m)$. As an example, the later result implies that every biplanar graph is 8-choosable with bounded clustering. This is the best known result for the clustered version of the earth-moon problem. The results extend to the setting where we only consider the maximum average degree of subgraphs with at least some number of vertices. Several applications are presented.

References in corpus (7)

• Improper Colourings inspired by Hadwiger's Conjecture
• A relative of Hadwiger's conjecture
• Defective colouring of graphs excluding a subgraph or minor
• Islands in minor-closed classes. I. Bounded treewidth and separators
• Islands in graphs on surfaces
• Improper coloring of graphs on surfaces
• A note on choosability with defect 1 of graphs on surfaces

Cited by in corpus (7)

• Clustered 3-Colouring Graphs of Bounded Degree
• Clustered Variants of Hajós' Conjecture
• Clustered Coloring of Graphs Excluding a Subgraph and a Minor
• Clustered Coloring of Graphs with Bounded Layered Treewidth and Bounded Degree
• Colouring Strong Products
• Defective DP-colorings of sparse multigraphs
• Defective DP-colorings of sparse simple graphs