Colouring and Covering Nowhere Dense Graphs

Abstract

In [12] it was shown that nowhere dense classes of graphs admit sparse neighbourhood covers of small degree. We show that a monotone graph class admits sparse neighbourhood covers if and only if it is nowhere dense. The existence of such covers for nowhere dense classes is established through bounds on so-called weak colouring numbers. The core results of this paper are various lower and upper bounds on the weak colouring numbers and other, closely related generalised colouring numbers. We prove tight bounds for these numbers on graphs of bounded treewidth. We clarify and tighten the relation between the density of shallow minors and the various generalised colouring numbers. These upper bounds are complemented by new, stronger exponential lower bounds on the weak and strong colouring numbers, and by super-polynomial lower bounds on the weak colouring numbers on classes of polynomial expansion. Finally, we show that computing weak r-colouring numbers is NP-complete for all r ≥ 3. Introduction.Nowhere dense classes of graphs have been introduced by Nešetřil and Ossona de Mendez [18,20] as a general model of uniformly sparse graph classes. They include and generalise many other natural sparse graph classes, among them all classes of bounded degree, classes of bounded genus, classes defined by excluded (topological) minors, and classes of bounded expansion. It has been demonstrated in several papers, e.g., [2,8,12,16,18] that nowhere dense graph classes have nice algorithmic properties; many problems that are hard in general can be solved (more) efficiently on nowhere dense graph classes. In fact, nowhere dense classes are a natural limit for the efficient solvability of a wide class of problems [7,12,15].In [12], it was shown that nowhere dense classes of graphs admit sparse neighbourhood covers. Neighbourhood covers play an important role in the study of distributed network algorithms and other application areas (see for example [21]). The neighbourhood covers developed in [12] combine low radius and low degree making them interesting for the applications outlined above. In this paper, we prove a (partial) converse to the result of [12]: we show that monotone graph classes (that is, classes closed under taking subgraphs) are nowhere dense if and only if they admit sparse neighbourhood covers. A similar characterisation result was proved for classes of bounded expansion [19].Nowhere denseness has turned out to be a very robust property of graph classes with various seemingly unrelated characterisations (see [11,18]), among them characterisations through so-called generalised colouring numbers. These are particularly relevant in the algorithmic context, because the existence of sparse neighbourhood covers for nowhere dense

On the generalised colouring numbers of graphs that exclude a fixed minor

“…More explicitly, for every graph G we have the following relations. The equality col ∞ (G) = tw(G) + 1 was first proved in [10]; for completeness we include the proof in Subsection 2.2. The equality wcol ∞ (G) = td(G) is proved in [22,Lemma 6.5].…”

“…• Lower bounds for the generalised colouring numbers for minor closed classes are given in [10]. In that paper it is shown that for every k and every r there is a graph G k,r of treewidth k that satisfies col r (G k,r ) = k + 1 and wcol r (G k,r ) = r+k k .…”

“…In Section 4 we prove Theorem 1.4 and in Section 5 we prove Theorem 1.2. Our proofs will rely on the notion of the elimination-width of a vertex-order < L , and its connection to weak colouring, stated as Theorem 2.1, which was proved in [10]. In Section 6 we prove Theorems 1.5 and 1.6, which have a detailed analysis of the generalised colouring numbers of planar graphs at their base.…”

“…Theorem 2.1 (Grohe et al [10]). Let G be a graph and let L ∈ Π(G) be a linear order of V (G) with elimination-width at most k. For all r ∈ N and all v ∈ V (G), we have…”

Polynomial bounds for centered colorings on proper minor-closed graph classes

“…We next show that if the considered tree decomposition has small adhesion, then every vertex reaches only a small number of vertices via short paths in the skeleton. The argument essentially boils down to the combinatorial core of the proof that graphs of treewidth k have weak p-coloring number O(p k ) [20]. Lemma 24.…”