Defective and Clustered Graph Colouring

David R. Wood

doi:10.37236/7406

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 Verdière parameter, graphs with given circumference, graphs excluding a given 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.

Highlights

Consider a graph where each vertex is assigned a colour
A graph G is k-colourable with defect d if each vertex of G can be assigned one of k colours such that each vertex is adjacent to at most d neighbours of the same colour; that is, each monochromatic component has maximum degree at most d
We prove by induction on |V (H)| + |E(H)| that every subgraph H of G is L-colourable with defect − k

Summary

Introduction

Consider a graph where each vertex is assigned a colour. A monochromatic component is a connected component of the subgraph induced by all the vertices assigned a single colour. A graph G is k-colourable with clustering c if each vertex can be assigned one of k colours such that each monochromatic component has at most c vertices. The emphasis is on general results for broadly defined classes of graphs, rather than more precise results for more specific classes With this viewpoint the following definitions naturally arise. The clustered chromatic number of a graph class G, denoted by χ (G), is the minimum integer k for which there exists an integer c such that every graph in G has a k-colouring with clustering c. A graph class G is defectively k-colourable if there exists an integer d such that every graph in G is k-colourable with defect d. The defective chromatic number of G, denoted by χ∆(G), is the minimum integer k such that G is defectively k-colourable.

History and Terminology

Definitions

Choosability

Standard Examples

Two Fundamental Observations