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Local convergence of random graph colorings

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Coja-Oghlan, Amin, Efthymiou, Charilaos and Jaafari, Nor (2018) Local convergence of random graph colorings. Combinatorica, 38 (2). pp. 341-380. doi:10.1007/s00493-016-3394-x ISSN 0209-9683.

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Official URL: https://doi.org/10.1007/s00493-016-3394-x

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Abstract

Let G(n,m) be a random graph whose average degree d=2m/n is below the k-colorability threshold. If we sample a k-coloring \SIGMA of G(n,m) uniformly at random, what can we say about the correlations between the colors assigned to vertices that are far apart?

According to a prediction from statistical physics, for average degrees below the so-called {\em condensation threshold} \dc, the colors assigned to far away vertices are asymptotically independent [Krzakala et al.: Proc. National Academy of Sciences 2007].

We prove this conjecture for k exceeding a certain constant k_0. More generally, we investigate the joint distribution of the k-colorings that \SIGMA induces locally on the bounded-depth neighborhoods of any fixed number of vertices. In addition, we point out an implication on the reconstruction problem.

Item Type: Journal Article
Subjects: Q Science > QA Mathematics
Divisions: Faculty of Science, Engineering and Medicine > Science > Computer Science
Library of Congress Subject Headings (LCSH): Graph coloring, Random graphs
Journal or Publication Title: Combinatorica
Publisher: Springer Berlin Heidelberg
ISSN: 0209-9683
Official Date: April 2018
Dates:
DateEvent
April 2018Published
13 June 2017Available
14 March 2016Accepted
Volume: 38
Number: 2
Page Range: pp. 341-380
DOI: 10.1007/s00493-016-3394-x
Status: Peer Reviewed
Publication Status: Published
Access rights to Published version: Restricted or Subscription Access
Date of first compliant deposit: 19 March 2019
Date of first compliant Open Access: 25 March 2019
RIOXX Funder/Project Grant:
Project/Grant IDRIOXX Funder NameFunder ID
278857–PTCCSeventh Framework Programmehttp://dx.doi.org/10.13039/100011102

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