Chasing the k-colorability threshold

Amin Coja-Oghlan, Dan Vilenchik

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

36 Scopus citations

Abstract

In this paper we establish a substantially improved lower bound on the k-colorability threshold of the random graph G(n,m) with n vertices and m edges. The new lower bound is ≈ 1.39 less than the 2k ln k-ln k first-moment upper bound (and ≈ 0.39 less than the 2k ln k - ln k - 1 physics conjecture). By comparison, the best previous bounds left a gap of about 2 + lnk, unbounded in terms of the number of colors [Achlioptas, Naor: STOC 2004]. Furthermore, we prove that, in a precise sense, our lower bound marks the so-called condensation phase transition predicted on the basis of physics arguments [Krzkala et al.: PNAS 2007]. Our proof technique is a novel approach to the second moment method, inspired by physics conjectures on the geometry of the set of k-colorings of the random graph.

Original languageEnglish
Title of host publicationProceedings - 2013 IEEE 54th Annual Symposium on Foundations of Computer Science, FOCS 2013
Pages380-389
Number of pages10
DOIs
StatePublished - 1 Dec 2013
Externally publishedYes
Event2013 IEEE 54th Annual Symposium on Foundations of Computer Science, FOCS 2013 - Berkeley, CA, United States
Duration: 27 Oct 201329 Oct 2013

Publication series

NameProceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS
ISSN (Print)0272-5428

Conference

Conference2013 IEEE 54th Annual Symposium on Foundations of Computer Science, FOCS 2013
Country/TerritoryUnited States
CityBerkeley, CA
Period27/10/1329/10/13

Keywords

  • Graph coloring
  • Phase transitions
  • Random structures

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