60 research outputs found
Disproof of the List Hadwiger Conjecture
The List Hadwiger Conjecture asserts that every -minor-free graph is
-choosable. We disprove this conjecture by constructing a
-minor-free graph that is not -choosable for every integer
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
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 Choosability of Sparse Graphs
An (improper) graph colouring has "defect" if each monochromatic subgraph
has maximum degree at most , and has "clustering" if each monochromatic
component has at most vertices. This paper studies defective and clustered
list-colourings for graphs with given maximum average degree. We prove that
every graph with maximum average degree less than is
-choosable with defect . This improves upon a similar result by Havet and
Sereni [J. Graph Theory, 2006]. For clustered choosability of graphs with
maximum average degree , no bound on the number of colours
was previously known. The above result with solves this problem. It
implies that every graph with maximum average degree is
-choosable with clustering 2. This extends a
result of Kopreski and Yu [Discrete Math., 2017] to the setting of
choosability. We then prove two results about clustered choosability that
explore the trade-off between the number of colours and the clustering. In
particular, we prove that every graph with maximum average degree is
-choosable with clustering , and is
-choosable with clustering . As an
example, the later result implies that every biplanar graph is 8-choosable with
bounded clustering. This is the best known result for the clustered version of
the earth-moon problem. The results extend to the setting where we only
consider the maximum average degree of subgraphs with at least some number of
vertices. Several applications are presented
Defective 3-Paintability of Planar Graphs
A -defective -painting game on a graph is played by two players:
Lister and Painter. Initially, each vertex is uncolored and has tokens. In
each round, Lister marks a chosen set of uncolored vertices and removes one
token from each marked vertex. In response, Painter colors vertices in a subset
of which induce a subgraph of maximum degree at most . Lister
wins the game if at the end of some round there is an uncolored vertex that has
no more tokens left. Otherwise, all vertices eventually get colored and Painter
wins the game. We say that is -defective -paintable if Painter has a
winning strategy in this game. In this paper we show that every planar graph is
3-defective 3-paintable and give a construction of a planar graph that is not
2-defective 3-paintable.Comment: 21 pages, 11 figure
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