Defective colouring of graphs
excluding a subgraph or minor
Archdeacon (1987) proved that graphs embeddable on a fixed surface can be -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 -minor can be -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 Verdière parameter, and graphs excluding a complete bipartite graph as a topological minor.
A graph is -colourable with defect , or -improper -colourable, or simply -colourable, if each vertex of can be assigned one of colours so that at most neighbours of are assigned the same colour as . That is, each monochromatic subgraph has maximum degree at most . Obviously the case corresponds to the usual notion of graph colouring. Cowen et al.  introduced the notion of defective graph colouring, and now many results for various graph classes are known. This paper presents -colourability results for graph classes defined by an excluded subgraph, subdivision or minor. Our primary focus is on minimising the number of colours rather than the degree bound . This viewpoint motivates the following definition. The defective chromatic number of a graph class is the minimum integer (if such a exists) for which there exists an integer such that every graph in is -colourable.
Consider the following two examples: Archdeacon  proved that for every surface , the defective chromatic number of graphs embeddable in equals 3. And Edwards, Kang, Kim, Oum, and Seymour  proved that the class of graphs containing no minor has defective chromatic number (which is a weakening of Hadwiger’s conjecture). This paper proves a general theorem that implies both these results as special cases. Indeed, our theorem only assumes an excluded subgraph, which enables it to be applied in more general settings.
For integers , let be the bipartite graph obtained from by adding new vertices, each adjacent to a distinct pair of vertices in the colour class of vertices in (see Figure 1). Our main result shows that every graph excluding as a subgraph is -colourable, where depends on , and certain density parameters, which we now introduce.
For a graph , the most natural density parameter to consider is the maximum average degree, denoted , which is the maximum of the average degrees of all subgraphs of ; that is,
Closely related to maximum average degree is degeneracy. A graph is -degenerate if every subgraph of has a vertex of degree at most . Every graph is -degenerate. It follows that the chromatic number (and even the choice number) of a graph is at most . Bounds on defective colourings have also been obtained in terms of maximum average degree. In particular, Havet and Sereni  proved that every graph with is -colourable, and that there exist non--colourable graphs whose maximum average degree tends to when goes to infinity, which shows the limit of the maximum average degree approach for defective colouring (see also [29, 10, 13, 52, 20, 15, 14, 24]).
In addition to maximum average degree we consider the density of shallow topological minors (see  for more on this topic). A graph is a minor of a graph if a graph isomorphic to can be obtained from a subgraph of by contracting edges. A graph is a topological minor of a graph if a subdivision of is a subgraph of . A -subdivision of a graph is a graph obtained from by subdividing each edge at most times, or equivalently replacing each edge by a path of length at most . The exact -subdivision of is the graph obtained from by subdividing each edge exactly once. For a half integer (that is, a number such that is an integer), a graph is a depth- topological minor of a graph if a -subdivision of is a subgraph of . For a graph , let be the set of all depth- topological minors of . The topological greatest reduced average density (or top-grad) with rank of a graph is defined as
Note that .
The following is our main result (see Theorem 2.3 for a more precise version).
Every graph with no subgraph has an -colouring, where depends on , , and
We actually prove this result in the setting of defective list colourings, introduced by Eaton and Hull  and since studied by several authors [17, 18, 19, 38, 61, 81, 87, 89, 94, 92, 93]. A -list assignment of a graph is a function that assigns a set of exactly colours to each vertex . Then an -colouring of is a function that assigns a colour in to each vertex . If an -colouring has the property that every vertex has at most neighbours having the same colour as , then we call it an -colouring with defect , or -defective. A graph is -choosable with defect , or -improper -choosable, or simply -choosable if for every -list assignment of , there exists a -defective -colouring of . For example, the result of Havet and Sereni  mentioned above applies in the setting of -choosability. The defective choice number of a graph class is the minimum integer (if such a exists) for which there exists an integer such that every graph in is -choosable.
The paper is organised as follows. Section 2 presents the proof of our main result (a choosability version of Theorem 1.1). The subsequent sections present several applications of this main result. In particular, Section 3 gives results for graphs with no 4-cycle, and other graph classes defined by an excluded subgraph. Section 4 presents defective 3-colourability results for graphs drawn on surfaces, even allowing for a linear number of crossings, thus generalising the result of Archdeacon mentioned above. Section 5 gives bounds on the defective chromatic number and defective choice number of graphs with given thickness, and of graphs with given stack- or queue-number. One result here is relevant to the earth–moon problem, which asks for the chromatic number of graphs with thickness 2. While it is open whether such graphs are 11-colourable, we prove they are 11-colourable with defect 2. Section 6 studies the defective chromatic number of minor-closed classes. We determine the defective chromatic number and defective choice number of linklessly embeddable graphs, knotlessly embeddable graphs, and graphs with given Colin de Verdière parameter. We then prove a strengthening of the result of Edwards et al.  mentioned above. Finally, we formulate a conjecture about the defective chromatic number of -minor-free graphs, and prove several special cases of it.
2. Main Proof
An edge in a graph is -light if both endpoints of have degree at most . There is a large literature on light edges in graphs; see [11, 49, 50, 47, 16, 9] for example. Many of our results rely on the following sufficient condition for -choosability. Its proof is essentially identical to the proof of a lemma by Lih et al. [61, Lemma 1]. Škrekovski  proved a similar result with .
For integers , if every subgraph of a graph has a vertex of degree at most or an -light edge, then is -choosable.
Let be a -list assignment for . We prove by induction on that every subgraph of is -colourable with defect . The base case with is trivial. Consider a subgraph of . If has a vertex of degree at most , then by induction is -colourable with defect , and there is a colour in used by no neighbour of which can be assigned to . Now assume that has minimum degree at least . By assumption, contains an -light edge . By induction, has an -colouring with defect . If , then is also an -colouring of with defect . Now assume that . We may further assume that is not an -colouring of with defect . Without loss of generality, has exactly neighbours (including ) coloured by . Since , there are at most neighbours not coloured by . Since contains colours different from , there is a colour used by no neighbour of which can be assigned to . ∎
To state our main result, we use the following auxiliary function. For positive integers , and positive reals and , let
For positive integers , , and positive reals , , let . If every subgraph of a graph has average degree at most and every graph whose exact -subdivision is a subgraph of has average degree at most , then at least one of the following holds:
contains a subgraph,
has a vertex of degree at most ,
has an -light edge.
The case is simple: If (i) does not hold, then , in which case either has no edges and (ii) holds, or has an edge and (iii) holds since . Now assume that .
Assume for contradiction that has no subgraph, that every vertex of has degree at least (thus ), and that contains no -light edge.
Let be the set of vertices in of degree at most . Let . Let and . Since has a vertex of degree at most and , we deduce that . Note that no two vertices in are adjacent.
Since the average degree of is at most ,
Let be a minor of obtained from by greedily finding a vertex having a pair of non-adjacent neighbours , in and replacing by an edge joining and (by deleting all edges incident with except , and contracting ), until no such vertex exists.
Let and . Clearly the exact -subdivision of is a subgraph of . So every subgraph of has average degree at most . Since contains at least edges,
A clique in a graph is a set of pairwise adjacent vertices. Let be the number of cliques of size in . Since is -degenerate,
For each vertex , since was not contracted in the creation of , the set of neighbours of in is a clique of size at least . Thus if , then there are at least vertices in sharing at least common neighbours in . These vertices and their common neighbours in with the vertices in form a subgraph of , contradicting our assumption. Thus,
contradicting the definition of . ∎
For integers , every graph with no subgraph is -choosable, where .
The following recursive construction was used by Edwards et al.  to show that their theorem mentioned above is tight. We use this example repeatedly, so include the proof for completeness. If , then let . If , then let be obtained from the disjoint union of copies of by adding one new vertex adjacent to all other vertices. Note that Havet and Sereni  used a similar construction to prove their lower bound mentioned above.
Lemma 2.4 (Edwards et al. ).
For integers and , the graph has no minor and no -colouring.
We proceed by induction on . In the base case, , and every 1-colouring has a colour class (the whole graph) inducing a subgraph with maximum degree larger than . Thus is not -colourable. Now assume that and the claim holds for . Let be the dominant vertex in . Let be the components of , where each is isomorphic to .
If contains a minor, then some component of contains a minor, which contradicts our inductive assumption. Thus contains no minor.
Suppose that is an -colouring of . We may assume that . If has a vertex of colour for each , then has more than neighbours of colour , which is not possible. Thus some component has no vertex coloured , and at most colours are used on . This contradicts the assumption that has no -colouring. ∎
Of course, for integers , the graph has no minor, no topological minor, and no -minor. Thus Lemma 2.4 shows that the number of colours in Theorem 2.3 is best possible. In other words, Theorem 2.3 states that defective chromatic number and defective choice number of every class of graphs of bounded with no subgraph are at most , and Lemma 2.4 shows that the number of colours cannot be decreased.
3. Excluded Subgraphs
This section presents several applications of our main result, in the setting of graph classes defined by an excluded subgraph. Since contains and a -subdivision of , Theorem 2.3 immediately implies:
Every graph with no subgraph is -choosable, where depends on , , and .
Every graph with no subgraph isomorphic to a -subdivision of is -choosable, where depends on , and .
3.1. No -Cycles
Since is the -cycle, Theorem 2.3 with and implies the following.
Every graph with no -cycle is -choosable, where .
Every graph with no minor nor -cycle subgraph is -choosable.
For planar graphs, . Indeed for planar graphs with at least three vertices. creftype 3.1 says that every planar graph with no -cycle is -choosable. However, better degree bounds are known. Borodin et al.  proved that every planar graph with no cycle of length has a vertex of degree at most or a -light edge. By Lemma 2.1, planar graphs with no -cycle are -choosable. Note that planar graphs with no -cycle are also known to be -choosable .
3.2. Edge Partitions
He et al.  proved the following theorem on partitioning a graph into edge-disjoint subgraphs.
Theorem 3.3 (He et al. [39, Theorem 3.1]).
If every subgraph of a graph has a vertex of degree at most or an -light edge, then has an edge-partition into two subgraphs and such that is a forest and is a graph with maximum degree at most .
This theorem and Lemma 2.2 imply the following.
For an integer , every graph with no subgraph has an edge-partition into two subgraphs and such that is a forest and has maximum degree at most .
3.3. Nowhere Dense Classes
A class of graphs is nowhere dense  if, for every integer there is some such that no -subdivision of is a subgraph of a graph in . Nowhere dense classes are also characterised  by the property that for every integer there exists a function with such that every graph of order in the class has . In other words, for each integer every graph in the class has .
For nowhere dense classes, there is no hope to find an improper colouring with a bounded number of colours, since the chromatic number of a nowhere dense class is typically unbounded (as witnessed by the class of graphs such that ). However, by the above characterisation, Theorem 2.3 implies there is a partition of the vertex set into a bounded number of parts, each with ‘small’ maximum degree.
Let be a nowhere dense class. Then there exist and a function with such that every -vertex graph in is -choosable.
4. 3-Colouring Graphs on Surfaces
This section considers defective colourings of graphs drawn on a surface, possibly with crossings. First consider the case of no crossings. For example, Cowen et al.  proved that every planar graph is -colourable, improved to -choosable by Eaton and Hull . Since is planar, by Lemma 2.4 the class of planar graphs has defective chromatic-number and defective choice number equal to 3. More generally, Archdeacon  proved the conjecture of Cowen et al.  that for every fixed surface , the class of graphs embeddable in has defective chromatic-number 3. Woodall  proved that such graphs have defective choice number 3. It follows from Euler’s formula that is not embeddable on for some constant (see Lemma 4.3), and that graphs embeddable in have bounded average degree and . Thus Theorem 2.3 implies Woodall’s result. The lower bound follows from Lemma 2.4 since is planar.
For every surface , the class of graphs embeddable in has defective chromatic-number 3 and defective choice number 3.
While our main goal is to bound the number of colours in a defective colouring, we now estimate the degree bound using our method for a graph embeddable in a surface of Euler genus . The Euler genus of an orientable surface with handles is . The Euler genus of a non-orientable surface with cross-caps is . The Euler genus of a graph is the minimum Euler genus of a surface in which embeds. For , define
The next two lemmas are well known. We include their proofs for completeness.
Every -vertex graph embeddable in a surface of Euler genus has at most edges.
Suppose that , where . We may assume that . By Euler’s Formula, , implying . Since we have . Since ,
Thus . By the quadratic formula, , which is a contradiction. Hence . ∎
Lemma 4.3 (Ringel ).
For every surface of Euler genus , the complete bipartite graph does not embed in .
By Euler’s formula, every triangle-free graph with vertices that embeds in has at most edges. The result follows. ∎
Theorems 2.3, 4.3 and 4.2 imply that graphs embeddable in are -choosable. This degree bound is weaker than the bound of obtained by Archdeacon. However, our bound is easily improved. Results by Jendro’l and Tuhársky  and Ivančo  show that every graph with Euler genus has a -light edge. Then Lemma 2.1 directly implies that every graph with Euler genus is -choosable. Still this bound is weaker than the subsequent improvements to Archdeacon’s result of -colourability by Cowen et al.  and to -choosability by Woodall ; also see .
4.1. Linear Crossing Number
We now generalise Theorem 4.1 to the setting of graphs with linear crossing number. For an integer and real number , say a graph is -close to Euler genus (resp. -close to planar) if every subgraph of has a drawing on a surface of Euler genus (resp. on the plane) with at most crossings. This says that the average number of crossings per edge is at most (for every subgraph). Of course, a graph is planar if and only if it is -close to planar, and a graph has Euler genus at most if and only if it is -close to Euler genus . Graphs that can be drawn in the plane with at most crossings per edge, so called -planar graphs, are examples of graphs -close to planar. Pach and Tóth  proved that -planar graphs have average degree . It follows that -planar graphs are -colourable, which is best possible since is -planar. For defective colourings, three colours suffice even in the more general setting of graphs -close to Euler genus .
For all integers the class of graphs -close to Euler genus has defective chromatic number and defective choice number equal to 3. In particular, every graph -close to Euler genus is -choosable.
We prove this theorem by a series of lemmas, starting with a straightforward extension of the standard probabilistic proof of the crossing lemma. Note that Shahrokhi et al.  obtained a better bound for a restricted range of values for relative to .
Every drawing of a graph with vertices and edges on a surface of Euler genus has at least crossings.
By Lemma 4.2, every -vertex graph that embeds in has at most edges. Thus every drawing of an -vertex -edge graph on has at least crossings.
Given a graph with vertices and edges and a crossing-minimal drawing of on , choose each vertex of independently and randomly with probability . Note that . Let be the induced subgraph obtained. The expected number of vertices in is , the expected number of edges in is , and the expected number of crossings in the induced drawing of is , where is the number of crossings in the drawing of . By linearity of expectation and the above naive bound, . Thus . ∎
This lemma leads to the following bound on the number of edges.
If an -vertex -edge graph has a drawing on a surface of Euler genus with at most crossings, then
If then , and we are done. Otherwise, , and Lemma 4.5 is applicable. Thus every drawing of on a surface of Euler genus has at least crossings. Hence , implying . ∎
To apply Theorem 2.3 we bound the size of subgraphs.
Every drawing of in a surface of Euler genus has at least
By Lemma 4.3, does not embed (crossing-free) in a surface of Euler genus . Consider a drawing of in a surface of Euler genus . There are copies of in . Each such copy has a crossing. Each crossing is in at most copies of . Thus the number of crossings is at least
If a graph is -close to Euler genus and contains as a subgraph, then
Suppose that contains as a subgraph. Since is -close to Euler genus , so is . Thus has a drawing in a surface of Euler genus where the number of crossings is at most . By Lemma 4.7,
The result follows. ∎
We now prove the main result of this section.
Proof of Theorem 4.4.
Say is a graph -close to Euler genus . By Lemma 4.8, contains no with . By Lemma 4.6, . We now bound . Consider a subgraph of that is a )-subdivision of a graph . Since is -close to Euler genus , so is . Thus has a drawing on a surface of Euler genus with at most crossings. Remove each division vertex and replace its two incident edges by one edge. We obtain a drawing of with the same number of crossings as the drawing of . Now . Thus has a drawing on a surface of Euler genus with at most crossings. By Lemma 4.6, . Hence . By Theorem 2.3, is -choosable, where
5. Thickness Parameters
This section studies defective colourings of graphs with given thickness or other related parameters. Yancey  first proposed studying defective colourings of graphs with given thickness.
5.1. Light Edge Lemma
Our starting point is the following sufficient condition for a graph to have a light edge. The proof uses a technique by Bose et al. , which we present in a general form.
Let be a graph with vertices, at most edges, and minimum degree , such that every spanning bipartite subgraph has at most edges, for some and and satisfying:
Then has an -light edge.
Let be the set of vertices with degree at most . Since vertices in have degree at least and vertices not in have degree at least ,
Suppose on the contrary that is an independent set in . Let be the spanning bipartite subgraph of consisting of all edges between and . Since each of the at least edges incident with each vertex in are in ,
Since (hence ) and ,
If then every edge is -light. Now assume that . Since ,
which is a contradiction. Thus is not an independent set. Hence contains an -light edge. ∎
To verify (6), the following approximation can be useful: If are strictly positive reals, then the larger root of is at most
Lemma 2.1 with and Lemma 5.1 imply the following sufficient condition for defective choosability. With , which is the minimum possible value for , the number of colours only depends on the coefficient of in the bound on the number of edges in a bipartite subgraph .
Lemma 5.1 with and and and and implies that every graph with minimum degree at least and Euler genus has a -light edge. Note that this bound is within of being tight since has minimum degree 3, embeds in a surface of Euler genus , and every edge has an endpoint of degree . More precise results, which are typically proved by discharging with respect to an embedding, are known [48, 44, 9]. Lemma 5.2 then implies that every graph with Euler genus is -choosable. As mentioned earlier, this result with a better degree bound was proved by Woodall ; also see . The utility of Lemma 5.2 is that it is immediately applicable in more general settings, as we now show.
The thickness of a graph is the minimum integer such that is the union of planar subgraphs; see  for a survey on thickness. A minimum-degree-greedy algorithm properly -colours a graph with thickness , and it is an open problem to improve this bound for . The result of Havet and Sereni  implies that graphs with thickness , which have maximum average degree less than , are -choosable, but gives no result with at most colours. We show below that graphs with thickness are -choosable, and that no result with at most colours is possible. That is, both the defective chromatic number and defective choice number of the class of graphs of thickness at most equal . In fact, the proof works in the following more general setting. For an integer , the -thickness of a graph is the minimum integer such that is the union of subgraphs each with Euler genus at most . This definition was implicitly introduced by Jackson and Ringel . By Euler’s Formula, every graph with vertices and -thickness has at most edges, and every spanning bipartite subgraph has at most edges. Lemma 5.1 with (using (7) to verify (6)) implies:
Every graph with minimum degree at least and -thickness at most has a -light edge.
We now determine the defective chromatic number and defective choice number of graphs with given -thickness.
For integers and , the class of graphs with -thickness at most has defective chromatic number and defective choice number equal to . In particular, every graph with -thickness at most is -choosable.
Lemmas 5.3 and 5.2 imply the upper bound. As usual, the lower bound is provided by . We now prove that has -thickness at most by induction on (with fixed). Note that is planar, and thus has -thickness 1. Let be the vertex of such that is the disjoint union of copies of . For , let be the vertex of the -th component of such that is the disjoint union of copies of . Let . Observe that each component of is isomorphic to and by induction, has -thickness at most . Since consists of copies of pasted on for some , is planar and thus has -thickness 1. Hence has -thickness at most . By Lemma 2.4, has no -colouring. Therefore the class of graphs with -thickness at most has defective chromatic number and defective choice number at least . ∎
The case and relates to the famous earth–moon problem [3, 35, 43, 73, 45], which asks for the maximum chromatic number of graphs with thickness . The answer is in . The result of Havet and Sereni  mentioned in Section 1 implies that graphs with thickness 2 are -choosable, -choosable, -choosable, -choosable, and -choosable because their maximum average degree is less than . But their result gives no bound with at most 6 colours. Theorem 5.4 says that the class of graphs with thickness 2 has defective chromatic number and defective choice number equal to . In particular, Lemma 5.2 implies that graphs with thickness 2 are -choosable, -choosable, -choosable, -choosable, -choosable, -choosable, and -choosable. This final result, which is also implied by the result of Havet and Sereni , is very close to the conjecture that graphs with thickness 2 are 11-colourable. Improving these degree bounds provides an approach for attacking the earth–moon problem.
5.3. Stack Layouts
A -stack layout of a graph consists of a linear ordering of and a partition of such that no two edges in cross with respect to for each . Here edges and cross if . A graph is a -stack graph if it has a -stack layout. The stack-number of a graph is the minimum integer for which is a -stack graph. Stack layouts are also called book embeddings, and stack-number is also called book-thickness, fixed outer-thickness and page-number. The maximum chromatic number of -stack graphs is in ; see . For defective colourings, colours suffice.
The class of -stack graphs has defective chromatic number and defective choice number equal to . In particular, every -stack graph is -choosable.
The lower bound follows from Lemma 2.4 since an easy inductive argument shows that is a -stack graph for all . For the upper bound, is not a -stack graph ; see also . Every -stack graph has average degree less than (see [51, 6, 30]) and (see ). The result follows from Theorem 2.3 with and , where . ∎
5.4. Queue Layouts
A -queue layout of a graph consists of a linear ordering of and a partition of such that no two edges in are nested with respect to for each . Here edges and are nested if . The queue-number of a graph is the minimum integer for which has a -queue layout. A graph is a -queue graph if it has a -queue layout. Dujmović and Wood  state that determining the maximum chromatic number of -queue graphs is an open problem, and showed lower and upper bounds of and . We provide the following partial answer to this question.
Every -queue graph is -choosable.
Consider the problem of partitioning the domain of a given poset into so that each has small poset dimension. The Hasse diagram of is the graph whose vertices are the elements of and whose edges correspond to the cover relation of . Here covers in if , and there is no element of such that , , and . A linear extension of is a total order on such that implies for every . The jump number of is the minimum number of consecutive elements of a linear extension of that are not comparable in , where the minimum is taken over all possible linear extensions of .
For every integer there is an integer such that the domain of any poset with jump number at most