{"title":"给定阶和大小的图中三角形的精确最小数目","authors":"Hong Liu, O. Pikhurko, Katherine Staden","doi":"10.1017/fmp.2020.7","DOIUrl":null,"url":null,"abstract":"What is the minimum number of triangles in a graph of given order and size? Motivated by earlier results of Mantel and Turán, Rademacher solved the first nontrivial case of this problem in 1941. The problem was revived by Erdős in 1955; it is now known as the Erdős–Rademacher problem. After attracting much attention, it was solved asymptotically in a major breakthrough by Razborov in 2008. In this paper, we provide an exact solution for all large graphs whose edge density is bounded away from $1$, which in this range confirms a conjecture of Lovász and Simonovits from 1975. Furthermore, we give a description of the extremal graphs.","PeriodicalId":2,"journal":{"name":"ACS Applied Bio Materials","volume":null,"pages":null},"PeriodicalIF":4.6000,"publicationDate":"2017-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/fmp.2020.7","citationCount":"19","resultStr":"{\"title\":\"THE EXACT MINIMUM NUMBER OF TRIANGLES IN GRAPHS WITH GIVEN ORDER AND SIZE\",\"authors\":\"Hong Liu, O. Pikhurko, Katherine Staden\",\"doi\":\"10.1017/fmp.2020.7\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"What is the minimum number of triangles in a graph of given order and size? Motivated by earlier results of Mantel and Turán, Rademacher solved the first nontrivial case of this problem in 1941. The problem was revived by Erdős in 1955; it is now known as the Erdős–Rademacher problem. After attracting much attention, it was solved asymptotically in a major breakthrough by Razborov in 2008. In this paper, we provide an exact solution for all large graphs whose edge density is bounded away from $1$, which in this range confirms a conjecture of Lovász and Simonovits from 1975. Furthermore, we give a description of the extremal graphs.\",\"PeriodicalId\":2,\"journal\":{\"name\":\"ACS Applied Bio Materials\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":4.6000,\"publicationDate\":\"2017-12-02\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://sci-hub-pdf.com/10.1017/fmp.2020.7\",\"citationCount\":\"19\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"ACS Applied Bio Materials\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1017/fmp.2020.7\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATERIALS SCIENCE, BIOMATERIALS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"ACS Applied Bio Materials","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1017/fmp.2020.7","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATERIALS SCIENCE, BIOMATERIALS","Score":null,"Total":0}
THE EXACT MINIMUM NUMBER OF TRIANGLES IN GRAPHS WITH GIVEN ORDER AND SIZE
What is the minimum number of triangles in a graph of given order and size? Motivated by earlier results of Mantel and Turán, Rademacher solved the first nontrivial case of this problem in 1941. The problem was revived by Erdős in 1955; it is now known as the Erdős–Rademacher problem. After attracting much attention, it was solved asymptotically in a major breakthrough by Razborov in 2008. In this paper, we provide an exact solution for all large graphs whose edge density is bounded away from $1$, which in this range confirms a conjecture of Lovász and Simonovits from 1975. Furthermore, we give a description of the extremal graphs.
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