47 research outputs found
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 -minor-free graph is 5-choosable. Both proofs rely on the
characterisation of -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
-minor, we argue that this proof suggests a possible approach for
attacking the Hadwiger Conjecture
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" if each monochromatic component has maximum degree at most
. A colouring has "clustering" 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\`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
as a minor, graphs excluding as a minor, and graphs excluding
an arbitrary graph 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