{"title":"关于弱Turán-good图","authors":"Dániel Gerbner","doi":"10.7151/dmgt.2510","DOIUrl":null,"url":null,"abstract":"Given graphs $H$ and $F$ with $\\chi(H)<\\chi(F)$, we say that $H$ is weakly $F$-Tur\\'an-good if among $n$-vertex $F$-free graphs, a $(\\chi(F)-1)$-partite graph contains the most copies of $H$. Let $H$ be a bipartite graph that contains a complete bipartite subgraph $K$ such that each vertex of $H$ is adjacent to a vertex of $K$. We show that $H$ is weakly $K_3$-Tur\\'an-good, improving a very recent asymptotic bound due to Grzesik, Gy\\H ori, Salia and Tompkins. They also showed that for any $r$ there exist graphs that are not weakly $K_r$-Tur\\'an-good. We show that for any non-bipartite $F$ there exists graphs that are not weakly $F$-Tur\\'an-good. We also show examples of graphs that are $C_{2k+1}$-Tur\\'an-good but not $C_{2\\ell+1}$-Tur\\'an-good for every $k>\\ell$.","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":"{\"title\":\"On weakly Turán-good graphs\",\"authors\":\"Dániel Gerbner\",\"doi\":\"10.7151/dmgt.2510\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Given graphs $H$ and $F$ with $\\\\chi(H)<\\\\chi(F)$, we say that $H$ is weakly $F$-Tur\\\\'an-good if among $n$-vertex $F$-free graphs, a $(\\\\chi(F)-1)$-partite graph contains the most copies of $H$. Let $H$ be a bipartite graph that contains a complete bipartite subgraph $K$ such that each vertex of $H$ is adjacent to a vertex of $K$. We show that $H$ is weakly $K_3$-Tur\\\\'an-good, improving a very recent asymptotic bound due to Grzesik, Gy\\\\H ori, Salia and Tompkins. They also showed that for any $r$ there exist graphs that are not weakly $K_r$-Tur\\\\'an-good. We show that for any non-bipartite $F$ there exists graphs that are not weakly $F$-Tur\\\\'an-good. We also show examples of graphs that are $C_{2k+1}$-Tur\\\\'an-good but not $C_{2\\\\ell+1}$-Tur\\\\'an-good for every $k>\\\\ell$.\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2022-07-25\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"2\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.7151/dmgt.2510\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.7151/dmgt.2510","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Given graphs $H$ and $F$ with $\chi(H)<\chi(F)$, we say that $H$ is weakly $F$-Tur\'an-good if among $n$-vertex $F$-free graphs, a $(\chi(F)-1)$-partite graph contains the most copies of $H$. Let $H$ be a bipartite graph that contains a complete bipartite subgraph $K$ such that each vertex of $H$ is adjacent to a vertex of $K$. We show that $H$ is weakly $K_3$-Tur\'an-good, improving a very recent asymptotic bound due to Grzesik, Gy\H ori, Salia and Tompkins. They also showed that for any $r$ there exist graphs that are not weakly $K_r$-Tur\'an-good. We show that for any non-bipartite $F$ there exists graphs that are not weakly $F$-Tur\'an-good. We also show examples of graphs that are $C_{2k+1}$-Tur\'an-good but not $C_{2\ell+1}$-Tur\'an-good for every $k>\ell$.