1,073 research outputs found
Sampling Colourings of the Triangular Lattice
We show that the Glauber dynamics on proper 9-colourings of the triangular
lattice is rapidly mixing, which allows for efficient sampling. Consequently,
there is a fully polynomial randomised approximation scheme (FPRAS) for
counting proper 9-colourings of the triangular lattice. Proper colourings
correspond to configurations in the zero-temperature anti-ferromagnetic Potts
model. We show that the spin system consisting of proper 9-colourings of the
triangular lattice has strong spatial mixing. This implies that there is a
unique infinite-volume Gibbs distribution, which is an important property
studied in statistical physics. Our results build on previous work by Goldberg,
Martin and Paterson, who showed similar results for 10 colours on the
triangular lattice. Their work was preceded by Salas and Sokal's 11-colour
result. Both proofs rely on computational assistance, and so does our 9-colour
proof. We have used a randomised heuristic to guide us towards rigourous
results.Comment: 42 pages. Added appendix that describes implementation. Added
ancillary file
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
A Proof of a Conjecture of Ohba
We prove a conjecture of Ohba which says that every graph on at most
vertices satisfies .Comment: 21 page
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
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