{"title":"The Bipartite Ramsey Numbers b(C2m; C2n)","authors":"Rui Zhang, Yongqi Sun, A. Wu","doi":"10.5281/ZENODO.1335404","DOIUrl":null,"url":null,"abstract":"Given bipartite graphs H1 and H2, the bipartite Ramsey number b(H1;H2) is the smallest integer b such that any subgraph G of the complete bipartite graph Kb,b, either G contains a copy of H1 or its complement relative to Kb,b contains a copy of H2. It is known that b(K2,2;K2,2) = 5, b(K2,3;K2,3) = 9, b(K2,4;K2,4) = 14 and b(K3,3;K3,3) = 17. In this paper we study the case that both H1 and H2 are even cycles, prove that b(C2m;C2n) ≥ m + n − 1 for m = n, and b(C2m;C6) = m+ 2 for m ≥ 4. Keywords—bipartite graph; Ramsey number; even cycle","PeriodicalId":23764,"journal":{"name":"World Academy of Science, Engineering and Technology, International Journal of Mathematical, Computational, Physical, Electrical and Computer Engineering","volume":"1 1","pages":"152-155"},"PeriodicalIF":0.0000,"publicationDate":"2013-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"13","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"World Academy of Science, Engineering and Technology, International Journal of Mathematical, Computational, Physical, Electrical and Computer Engineering","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.5281/ZENODO.1335404","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 13
Abstract
Given bipartite graphs H1 and H2, the bipartite Ramsey number b(H1;H2) is the smallest integer b such that any subgraph G of the complete bipartite graph Kb,b, either G contains a copy of H1 or its complement relative to Kb,b contains a copy of H2. It is known that b(K2,2;K2,2) = 5, b(K2,3;K2,3) = 9, b(K2,4;K2,4) = 14 and b(K3,3;K3,3) = 17. In this paper we study the case that both H1 and H2 are even cycles, prove that b(C2m;C2n) ≥ m + n − 1 for m = n, and b(C2m;C6) = m+ 2 for m ≥ 4. Keywords—bipartite graph; Ramsey number; even cycle
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