Defective and Clustered Graph Colouring∗ † David R. Wood Abstract. defect d Consider the following two ways to colour the verticesd of a graph whereclustering the crequirement that adjacent vertices get distinct coloursc is relaxed. A colouring has if each monochromatic component has maximum degree at most . A colouring has if each monochromatic component has at most 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`ereparameter,Kt graphs with givenK circumference,s;t graphs excluding a fixed graph as an immersion,H graphs with given thickness, graphs with given stack- or queue-number, graphs excluding as a minor, graphs excluding as a minor, and graphs excluding an arbitrary graph as a minor. Several open problems are discussed. arXiv:1803.07694v1 [math.CO] 20 Mar 2018 ∗ Electronic Journal of Combinatorics c Davidhttp://www.combinatorics.org/DS23 R. Wood. Released under the CC BY license (International 4.0). † This is a preliminary version of a dynamic survey to be published in the [email protected] , #DS23, . School of Mathematical Sciences, Monash University, Melbourne, Australia ( ). Research supported by the Australian Research Council. 1 Contents 1 Introduction3 1.1 History and Terminology . .3 1.2 Definitions . .5 1.3 Choosability . .6 1.4 Standard Examples . .6 1.5 Two Fundamental Observations . .8 2 Greedy Approaches 10 1.6 Related Topics . .9 2.1 Light Edges . 10 3 Graphs on Surfaces 11 2.2 Islands . 10 3.1 Outerplanar Graphs . 11 3.2 Planar Graphs . 11 3.3 Hex Lemma . 13 3.4 Defective Colouring of Graphs on Surfaces . 14 4 Maximum Degree 19 3.5 Clustered Colouring of Graphs on Surfaces . 16 5 Maximum Average Degree 23 6 Excluding a Subgraph 25 7 Excluding a Shallow Minor 26 ∗ Ks;t 7.1 Excluding ...................................... 26 7.2 Linklessly Embeddable Graphs . 28 7.3 Knotlessly Embeddable Graphs . 29 7.4 Colin de Verdi`ereParameter . 30 7.5 Crossings . 31 7.6 Stack and Queue Layouts . 32 8 Minor-Closed Classes 36 7.7 Excluded Immersions . 33 Kt 8.1 ExcludingH a Minor and Bounded Degree . 36 8.2 -Minor-Free Graphs . 38 8.3 -Minor-Free Graphs . 41 8.4 Conjectures . 44 9 Thickness 48 8.5 Circumference . 46 9.1 Defective Colouring . 49 10 General Setting 52 9.2 Clustered Colouring . 51 2 1 Introduction monochromatic component ConsiderG a graphk where each vertexclustering is assignedc a colour. A k is a connected component of the subgraph induced by all thec vertices assignedG a singlek colour. A graphdefectisd -colourable withG if each vertexk can be assigned one of colours such that each monochromaticd component has at most vertices. A graph is -colourable with if each vertex ofd can be assigned one of colours such that each vertex is adjacent to at most neighbours of the same colour; that is, each monochromatic component has maximum degree at most . This paper surveys results and open problems regarding clustered and defective graph colouring, where the first priority is to minimise the number of colours, with small defect or small clustering as a secondary goal. We include various proofs that highlight the main methods employed. The emphasisclustered is on chromatic generalresults number for broadly defined classes of graphs,χ?( ) rather than more precisek G G results for more specific classes.c With this viewpoint the followingk definitions naturally arise. G c k unbounded The of a graphG class , denoted by , is the minimum integer defectively k d for whichG there exists an integer such that every graph in has a -colouring with clusteringG k d defective chromatic number χ∆( ) . If there is no such integer , then has clusteredG chromatic number.G A graph k k k class is -colourableG if there exists an integer such that every graph in is unbounded c G-colourable with defect . The of , denoted by , is the c 1 χ∆( ) χ?( ) χ( ) minimum integer− such thatG 6is defectivelyG 6 G -colourable. If thereG is no such integer , then has defective chromatic number. Every colouring of a graph with clustering has defect . Thus for every class . Tables1 and2 summarise 1.1the results History presented and Terminology in this survey; see Sections 1.2 and 1.3 for the relevant definitions. There is no single origin for the notions of defective and clustered graph colouring, and the terminology used in the literature is inconsistent.k d (k; d)- Earlycolourable papers on defective colouring include [16, 17, 128, 138, 166, 212], although these did notd-relaxed use the termd-improper ‘defect’. The definition of “ -colourable with defect ”, often written G, was introducedd by Cowen et al. [d66-improper]. This terminology chromatic number is fairly standard,d-chromatic although number or G is sometimes used. The minimumdefective number chromatic of colours number in a colouring of a graph with defect has been called the [144] or [16] of . Cowen et al. [65] introduced the of a graph class (as defined above). (k; c)-fragmented Alsok for clustered colouring, thec literature is inconsistent. Onec of the early papers is by Kleinberg et al. [151], who defined a colouring to be if, in our language, it is a -colouring with clusteringchromon. I prefer “clustering” since as increases, intuitively the “fragmentation”metachromatic of the number monochromatic components decreases. Edwards and Farr [89, 90] called a monochromatic component a and called the clustered chromatic number of a class the . 3 ` χ∆( ) χ ( ) G G ∆ G Table 1: Summary of Results for Defective Colouring O graph class Sect. P outerplanar 6 g g 2 2 3.1 E m m m m + 1 + 1 planar 6 A b 2 c3b 2 c 3 3.2 4 4 Euler genus L 3 3 3.4 5 5 max average degree K 5 k k k k linklessly embeddable6 V 7.2 k k k + 1 k + 1 knotlessly embeddable S 7.3 k k k + 1;:::; 2k + 1 2k + 1 Colin de Verdi`ere Q 7.4 Kt t 2 t 1 -stack graphs I − 7.6 -queue graphs6 k k + 1 k + 1 7.6 no immersion6 k 6 ∆ 2 2 7.7 treewidthKt Kt t 1 t 1 8.1 M − td(H)+1 − treewidthH , max degree H td(H) 1;:::; 2 4 min s : t H Ks;t 8.1 M − − f 9 g no Ks;t-minor (s 6 t Ks;t s s 8.2 M k+1 no -minor 6 k k log2 k + 1;:::; 3 log2 k 2 8.3 C b c b c d k+1 e no (k +-minor 1) ) k log2(k + 2) 1;:::; 3 log2 k 2 8.4 Hg d e − b c b c circumferenceg k 2k + 1 2k + 1 8.5 6 Tk no -path 8.5 -thickness 9 ` χ?( ) χ ( ) G G ? G Table 2: SummaryO of Results for Clustered Colouring graph class Sect. P g g outerplanar 6 E 3 3 3.1 planar 6 g 6 ∆ 43 3 4; 4 3.2 ∆+6 ∆+1 ∆+6 Euler genus6 ∆ ∆ 4 ;:::;43 4 ;:::; 4 ∆ + 1 3.5 D m m Euler genus , max degreem m + 1;:::; m + 1 + 1;:::; m + 1 8.1 6 A b 2 c b c b 2 c b c 5 5 max degree L 4 6 6 max average degree K 5 k k linklessly embeddable6 V 7.2 k k k + 2;:::; 2k + 2 k + 2 ::: 2k + 2 knotlessly embeddable S 7.3 Colink de Verdi`ere k k +open 1;:::; 4k k + open1;:::; 4k 7.4 Q t+4 Kt t t 1 -stack graphs I > b 4 c > − 7.6 -queue graphs6 k k + 1 k + 1 7.6 no immersion6 k 6 ∆ 2 2 7.7 31 Kt K t 1;:::; 2t 2 t 1;:::; t treewidth M t − − − d 2 e 8.1 treewidthKt , max degree6 ∆ 3 > 3 8.1 td(H)+1 no H -minor H td(H) 1;:::; 2 4 8.2 M − − no Ks;t-minor, max(s degreet K s + 1;:::; 2s + 2 8.2 6 M s;t k k log k + 1;:::; 3 log k no -minor 6 C b 2 c b 2 c open 8.3 no (k +-minor 1) ) k log2(k + 2) 1;:::; 3 log2 k open 8.4 Hg d e − b c g k 2k + 2;:::; 6k + 1 2k + 2;:::; 6k + 1 circumference6 Tk open 8.5 no -path open 8.5 -thickness 9 4 1.2 Definitions clique This section briefly states standard graph theoretic definitions, familiar to most readers. G k-colouring G k A G in a graphvw isG a setbichromatic of pairwise adjacentv w vertices. v LetG properlybe a graph. A v of is a function that assigns one of coloursv to each vertex of G. Anproper edge of is if and are assignedchromatic distinct number colours.G A vertexχ(Gof) is coloured ifk is assigned a colour distinctk from every neighbourG of . A colouring of graphis parameterif every vertex is properly coloured.f The of ,f denoted(G1) = f(G2), is the minimum integerG1 suchG2 that there is a properf -colouringbounded of . G c f(G) c G f unbounded f A is a real-valued6 function2 G on the class of graphs such thatG f( ) := sup f(G): G wheneverG graphsG and f are isomorphic.2 Gg Say is on a graph class if there f f( ) = max f(G): G exists such that for every , otherwiseG is G onf .