647 research outputs found
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
Deterministic counting of graph colourings using sequences of subgraphs
In this paper we propose a deterministic algorithm for approximately counting
the -colourings of sparse random graphs . In particular, our
algorithm computes in polynomial time a approximation of
the logarithm of the number of -colourings of for with high probability over the graph instances.
Our algorithm is related to the algorithms of A. Bandyopadhyay et al. in SODA
'06, and A. Montanari et al. in SODA '06, i.e. it uses {\em spatial correlation
decay} to compute {\em deterministically} marginals of {\em Gibbs
distribution}. We develop a scheme whose accuracy depends on {\em
non-reconstruction} of the colourings of , rather than {\em
uniqueness} that are required in previous works. This leaves open the
possibility for our schema to be sufficiently accurate even for .
The set up for establishing correlation decay is as follows: Given
, we alter the graph structure in some specific region of
the graph by deleting edges between vertices of . Then we show that
the effect of this change on the marginals of Gibbs distribution, diminishes as
we move away from . Our approach is novel and suggests a new context
for the study of deterministic counting algorithms
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
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