{"title":"m-二分Ramsey数BRm（H1，H2）","authors":"Yaser Rowshan","doi":"10.7151/dmgt.2477","DOIUrl":null,"url":null,"abstract":"Abstract In a (G1, G2) coloring of a graph G, every edge of G is in G1 or G2. For two bipartite graphs H1 and H2, the bipartite Ramsey number BR(H1, H2) is the least integer b ≥ 1, such that for every (G1, G2) coloring of the complete bipartite graph Kb,b, results in either H1 ⊆ G1 or H2 ⊆ G2. As another view, for bipartite graphs H1 and H2 and a positive integer m, the m-bipartite Ramsey number BRm(H1, H2) of H1 and H2 is the least integer n (n ≥ m) such that every subgraph G of Km,n results in H1 ⊆ G or H2 ⊆ Ḡ. The size of m-bipartite Ramsey number BRm(K2,2, K2,2), the size of m-bipartite Ramsey number BRm(K2,2, K3,3) and the size of m-bipartite Ramsey number BRm(K3,3, K3,3) have been computed in several articles up to now. In this paper we determine the exact value of BRm(K2,2, K4,4) for each m ≥ 2.","PeriodicalId":48875,"journal":{"name":"Discussiones Mathematicae Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.5000,"publicationDate":"2022-12-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":"{\"title\":\"The m-Bipartite Ramsey Number BRm(H1, H2)\",\"authors\":\"Yaser Rowshan\",\"doi\":\"10.7151/dmgt.2477\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Abstract In a (G1, G2) coloring of a graph G, every edge of G is in G1 or G2. For two bipartite graphs H1 and H2, the bipartite Ramsey number BR(H1, H2) is the least integer b ≥ 1, such that for every (G1, G2) coloring of the complete bipartite graph Kb,b, results in either H1 ⊆ G1 or H2 ⊆ G2. As another view, for bipartite graphs H1 and H2 and a positive integer m, the m-bipartite Ramsey number BRm(H1, H2) of H1 and H2 is the least integer n (n ≥ m) such that every subgraph G of Km,n results in H1 ⊆ G or H2 ⊆ Ḡ. The size of m-bipartite Ramsey number BRm(K2,2, K2,2), the size of m-bipartite Ramsey number BRm(K2,2, K3,3) and the size of m-bipartite Ramsey number BRm(K3,3, K3,3) have been computed in several articles up to now. In this paper we determine the exact value of BRm(K2,2, K4,4) for each m ≥ 2.\",\"PeriodicalId\":48875,\"journal\":{\"name\":\"Discussiones Mathematicae Graph Theory\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.5000,\"publicationDate\":\"2022-12-24\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"2\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Discussiones Mathematicae Graph Theory\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.7151/dmgt.2477\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Discussiones Mathematicae Graph Theory","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.7151/dmgt.2477","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
Abstract In a (G1, G2) coloring of a graph G, every edge of G is in G1 or G2. For two bipartite graphs H1 and H2, the bipartite Ramsey number BR(H1, H2) is the least integer b ≥ 1, such that for every (G1, G2) coloring of the complete bipartite graph Kb,b, results in either H1 ⊆ G1 or H2 ⊆ G2. As another view, for bipartite graphs H1 and H2 and a positive integer m, the m-bipartite Ramsey number BRm(H1, H2) of H1 and H2 is the least integer n (n ≥ m) such that every subgraph G of Km,n results in H1 ⊆ G or H2 ⊆ Ḡ. The size of m-bipartite Ramsey number BRm(K2,2, K2,2), the size of m-bipartite Ramsey number BRm(K2,2, K3,3) and the size of m-bipartite Ramsey number BRm(K3,3, K3,3) have been computed in several articles up to now. In this paper we determine the exact value of BRm(K2,2, K4,4) for each m ≥ 2.
期刊介绍：
The Discussiones Mathematicae Graph Theory publishes high-quality refereed original papers. Occasionally, very authoritative expository survey articles and notes of exceptional value can be published. The journal is mainly devoted to the following topics in Graph Theory: colourings, partitions (general colourings), hereditary properties, independence and domination, structures in graphs (sets, paths, cycles, etc.), local properties, products of graphs as well as graph algorithms related to these topics.
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