{"title":"Maximum independent sets on random regular graphs","authors":"Jian Ding, A. Sly, Nike Sun","doi":"10.14288/1.0044424","DOIUrl":null,"url":null,"abstract":"We determine the asymptotics of the independence number of the random d-regular graph for all $${d\\geq d_0}$$d≥d0. It is highly concentrated, with constant-order fluctuations around $${n\\alpha_*-c_*\\log n}$$nα∗-c∗logn for explicit constants $${\\alpha_*(d)}$$α∗(d) and $${c_*(d)}$$c∗(d). Our proof rigorously confirms the one-step replica symmetry breaking heuristics for this problem, and we believe the techniques will be more broadly applicable to the study of other combinatorial properties of random graphs.","PeriodicalId":50895,"journal":{"name":"Acta Mathematica","volume":null,"pages":null},"PeriodicalIF":4.9000,"publicationDate":"2013-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"60","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Acta Mathematica","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.14288/1.0044424","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 60
Abstract
We determine the asymptotics of the independence number of the random d-regular graph for all $${d\geq d_0}$$d≥d0. It is highly concentrated, with constant-order fluctuations around $${n\alpha_*-c_*\log n}$$nα∗-c∗logn for explicit constants $${\alpha_*(d)}$$α∗(d) and $${c_*(d)}$$c∗(d). Our proof rigorously confirms the one-step replica symmetry breaking heuristics for this problem, and we believe the techniques will be more broadly applicable to the study of other combinatorial properties of random graphs.