Defective and Clustered Graph Colouring

TL;DR

This survey reviews research on defective and clustered graph colourings, focusing on minimizing colours with small defect or clustering across various graph classes and variants.

Contribution

It compiles and discusses recent advances and open problems in defective and clustered colourings for numerous graph classes and variants.

Findings

Various bounds established for defective and clustered colourings.

Identification of open problems in the field.

Analysis of list colouring variants.

Abstract

Consider the following two ways to colour the vertices of a graph where the requirement that adjacent vertices get distinct colours is relaxed. A colouring has "defect" $d$ if each monochromatic component has maximum degree at most $d$. A colouring has "clustering" $c$ if each monochromatic component has at most $c$ vertices. This paper surveys research on these types of colourings, where the first priority is to minimise the number of colours, with small defect or small clustering as a secondary goal. List colouring variants are also considered. The following graph classes are studied: outerplanar graphs, planar graphs, graphs embeddable in surfaces, graphs with given maximum degree, graphs with given maximum average degree, graphs excluding a given subgraph, graphs with linear crossing number, linklessly or knotlessly embeddable graphs, graphs with given Colin de Verdi\`ere parameter, graphs with given circumference, graphs excluding a fixed graph as an immersion, graphs with given thickness, graphs with given stack- or queue-number, graphs excluding $K_t$ as a minor, graphs excluding $K_{s,t}$ as a minor, and graphs excluding an arbitrary graph $H$ as a minor. Several open problems are discussed.