Discrete Mathematics & Theoretical Computer Science, Vol 8 (2006)

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DMTCS vol 8 no 1 (2006), pp. 121-128

DMTCS

Volume 8

n° 1 (2006), pp. 121-128

author:Alan Frieze and Juan Vera
title:On randomly colouring locally sparse graphs
keywords:Counting Colourings, Sampling, Markov Chains
abstract:We consider the problem of generating a random
q
-colouring of a graph
G=(V,E)
. We consider the simple Glauber Dynamics chain. We show that if for all
v ∈ V
the average degree of the subgraph
H
v
induced by the neighbours of
v ∈ V
is
≪Δ
where
Δ
is the maximum degree and
Δ>c
1
ln
n
then for sufficiently large
c
1
, this chain mixes rapidly provided
q/Δ>α
, where
α≈ 1.763
is the root of
α = e
{1/α}
. For this class of graphs, which includes planar graphs, triangle free graphs and random graphs
G
{n,p}
with
p ≪ 1
, this beats the
11Δ/6
bound of Vigoda for general graphs.
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reference: Alan Frieze and Juan Vera (2006), On randomly colouring locally sparse graphs, Discrete Mathematics and Theoretical Computer Science 8, pp. 121-128
bibtex:For a corresponding BibTeX entry, please consider our BibTeX-file.
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