Thomassen's Choosability Argument Revisited

Thomassen (1994) proved that every planar graph is 5-choosable. This result was generalised by {\v{S}}krekovski (1998) and He et al. (2008), who proved that every $K_5$-minor-free graph is 5-choosable. Both proofs rely on the characterisation of $K_5$-minor-free graphs due to Wagner (1937). This paper proves the same result without using Wagner's structure theorem or even planar embeddings. Given that there is no structure theorem for graphs with no $K_6$-minor, we argue that this proof suggests a possible approach for attacking the Hadwiger Conjecture

Disproof of the List Hadwiger Conjecture

The List Hadwiger Conjecture asserts that every $K_t$-minor-free graph is $t$-choosable. We disprove this conjecture by constructing a $K_{3t+2}$-minor-free graph that is not $4t$-choosable for every integer $t\geq 1$

Flexible List Colorings in Graphs with Special Degeneracy Conditions

For a given $\varepsilon > 0$, we say that a graph $G$ is $\varepsilon$-flexibly $k$-choosable if the following holds: for any assignment $L$ of color lists of size $k$ on $V(G)$, if a preferred color from a list is requested at any set $R$ of vertices, then at least $\varepsilon |R|$ of these requests are satisfied by some $L$-coloring. We consider the question of flexible choosability in several graph classes with certain degeneracy conditions. We characterize the graphs of maximum degree $\Delta$ that are $\varepsilon$-flexibly $\Delta$-choosable for some $\varepsilon = \varepsilon(\Delta) > 0$, which answers a question of Dvo\v{r}\'ak, Norin, and Postle [List coloring with requests, JGT 2019]. In particular, we show that for any $\Delta\geq 3$, any graph of maximum degree $\Delta$ that is not isomorphic to $K_{\Delta+1}$ is $\frac{1}{6\Delta}$-flexibly $\Delta$-choosable. Our fraction of $\frac{1}{6 \Delta}$ is within a constant factor of being the best possible. We also show that graphs of treewidth $2$ are $\frac{1}{3}$-flexibly $3$-choosable, answering a question of Choi et al.~[arXiv 2020], and we give conditions for list assignments by which graphs of treewidth $k$ are $\frac{1}{k+1}$-flexibly $(k+1)$-choosable. We show furthermore that graphs of treedepth $k$ are $\frac{1}{k}$-flexibly $k$-choosable. Finally, we introduce a notion of flexible degeneracy, which strengthens flexible choosability, and we show that apart from a well-understood class of exceptions, 3-connected non-regular graphs of maximum degree $\Delta$ are flexibly $(\Delta - 1)$-degenerate.Comment: 21 pages, 5 figure

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

Improper choosability and Property B

A fundamental connection between list vertex colourings of graphs and Property B (also known as hypergraph 2-colourability) was already known to Erd\H{o}s, Rubin and Taylor. In this article, we draw similar connections for improper list colourings. This extends results of Kostochka, Alon, and Kr\'al' and Sgall for, respectively, multipartite graphs, graphs of large minimum degree, and list assignments with bounded list union.Comment: 12 page