{"title":"The Freezing Threshold for k-Colourings of a Random Graph","authors":"Michael Molloy","doi":"10.1145/3034781","DOIUrl":null,"url":null,"abstract":"We determine the exact value of the freezing threshold, rfk, for k-colourings of a random graph when k≥ 14. We prove that for random graphs with density above rfk, almost every colouring is such that a linear number of vertices are frozen, meaning that their colour cannot be changed by a sequence of alterations whereby we change the colours of o(n) vertices at a time, always obtaining another proper colouring. When the density is below rfk, then almost every colouring is such that every vertex can be changed by a sequence of alterations where we change O(log n) vertices at a time. Frozen vertices are a key part of the clustering phenomena discovered using methods from statistical physics. The value of the freezing threshold was previously determined by the nonrigorous cavity method.","PeriodicalId":17199,"journal":{"name":"Journal of the ACM (JACM)","volume":"34 1","pages":"1 - 62"},"PeriodicalIF":0.0000,"publicationDate":"2018-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"38","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of the ACM (JACM)","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1145/3034781","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 38
Abstract
We determine the exact value of the freezing threshold, rfk, for k-colourings of a random graph when k≥ 14. We prove that for random graphs with density above rfk, almost every colouring is such that a linear number of vertices are frozen, meaning that their colour cannot be changed by a sequence of alterations whereby we change the colours of o(n) vertices at a time, always obtaining another proper colouring. When the density is below rfk, then almost every colouring is such that every vertex can be changed by a sequence of alterations where we change O(log n) vertices at a time. Frozen vertices are a key part of the clustering phenomena discovered using methods from statistical physics. The value of the freezing threshold was previously determined by the nonrigorous cavity method.