Defective and Clustered Graph Colouring

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.Comment: This is a preliminary version of a dynamic survey to be published in the Electronic Journal of Combinatoric

Extremal density for sparse minors and subdivisions

We prove an asymptotically tight bound on the extremal density guaranteeing subdivisions of bounded-degree bipartite graphs with a mild separability condition. As corollaries, we answer several questions of Reed and Wood on embedding sparse minors. Among others, $\bullet$ $(1+o(1))t^2$ average degree is sufficient to force the $t\times t$ grid as a topological minor; $\bullet$ $(3/2+o(1))t$ average degree forces every $t$-vertex planar graph as a minor, and the constant $3/2$ is optimal, furthermore, surprisingly, the value is the same for $t$-vertex graphs embeddable on any fixed surface; $\bullet$ a universal bound of $(2+o(1))t$ on average degree forcing every $t$-vertex graph in any nontrivial minor-closed family as a minor, and the constant 2 is best possible by considering graphs with given treewidth.Comment: 33 pages, 6 figure

Product structure of graph classes with bounded treewidth

We show that many graphs with bounded treewidth can be described as subgraphs of the strong product of a graph with smaller treewidth and a bounded-size complete graph. To this end, define the "underlying treewidth" of a graph class $\mathcal{G}$ to be the minimum non-negative integer $c$ such that, for some function $f$, for every graph ${G \in \mathcal{G}}$ there is a graph $H$ with ${\text{tw}(H) \leq c}$ such that $G$ is isomorphic to a subgraph of ${H \boxtimes K_{f(\text{tw}(G))}}$. We introduce disjointed coverings of graphs and show they determine the underlying treewidth of any graph class. Using this result, we prove that the class of planar graphs has underlying treewidth 3; the class of $K_{s,t}$-minor-free graphs has underlying treewidth $s$ (for ${t \geq \max\{s,3\}}$); and the class of $K_t$-minor-free graphs has underlying treewidth ${t-2}$. In general, we prove that a monotone class has bounded underlying treewidth if and only if it excludes some fixed topological minor. We also study the underlying treewidth of graph classes defined by an excluded subgraph or excluded induced subgraph. We show that the class of graphs with no $H$ subgraph has bounded underlying treewidth if and only if every component of $H$ is a subdivided star, and that the class of graphs with no induced $H$ subgraph has bounded underlying treewidth if and only if every component of $H$ is a star

Graph Theory

[no abstract available