Defective colouring of graphs excluding a subgraph or minor

TL;DR

This paper generalizes defective coloring theorems for graphs excluding certain subgraphs or minors, applying to various graph classes such as those with bounded crossing number, thickness, or embeddability properties.

Contribution

It introduces a unified theorem that weakens previous assumptions and extends defective coloring results to broader classes of graphs.

Findings

Graphs with linear crossing number are defective 3-colorable.

Graphs with bounded thickness admit defective colorings.

Graphs excluding certain topological minors can be defectively colored.

Abstract

Archdeacon (1987) proved that graphs embeddable on a fixed surface can be $3$-coloured so that each colour class induces a subgraph of bounded maximum degree. Edwards, Kang, Kim, Oum and Seymour (2015) proved that graphs with no $K_{t+1}$-minor can be $t$-coloured so that each colour class induces a subgraph of bounded maximum degree. We prove a common generalisation of these theorems with a weaker assumption about excluded subgraphs. This result leads to new defective colouring results for several graph classes, including graphs with linear crossing number, graphs with given thickness (with relevance to the earth-moon problem), graphs with given stack- or queue-number, linklessly or knotlessly embeddable graphs, graphs with given Colin de Verdi\`ere parameter, and graphs excluding a complete bipartite graph as a topological minor.