邦迪泛周期定理的一般化

Nemanja Draganić, David Munhá Correia, Benny Sudakov
{"title":"邦迪泛周期定理的一般化","authors":"Nemanja Draganić, David Munhá Correia, Benny Sudakov","doi":"10.1017/s0963548324000075","DOIUrl":null,"url":null,"abstract":"The <jats:italic>bipartite independence number</jats:italic> of a graph <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000075_inline1.png\" /> <jats:tex-math> $G$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, denoted as <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000075_inline2.png\" /> <jats:tex-math> $\\tilde \\alpha (G)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, is the minimal number <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000075_inline3.png\" /> <jats:tex-math> $k$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> such that there exist positive integers <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000075_inline4.png\" /> <jats:tex-math> $a$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000075_inline5.png\" /> <jats:tex-math> $b$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> with <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000075_inline6.png\" /> <jats:tex-math> $a+b=k+1$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> with the property that for any two disjoint sets <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000075_inline7.png\" /> <jats:tex-math> $A,B\\subseteq V(G)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> with <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000075_inline8.png\" /> <jats:tex-math> $|A|=a$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000075_inline9.png\" /> <jats:tex-math> $|B|=b$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, there is an edge between <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000075_inline10.png\" /> <jats:tex-math> $A$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000075_inline11.png\" /> <jats:tex-math> $B$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. McDiarmid and Yolov showed that if <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000075_inline12.png\" /> <jats:tex-math> $\\delta (G)\\geq \\tilde \\alpha (G)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> then <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000075_inline13.png\" /> <jats:tex-math> $G$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is Hamiltonian, extending the famous theorem of Dirac which states that if <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000075_inline14.png\" /> <jats:tex-math> $\\delta (G)\\geq |G|/2$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> then <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000075_inline15.png\" /> <jats:tex-math> $G$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is Hamiltonian. In 1973, Bondy showed that, unless <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000075_inline16.png\" /> <jats:tex-math> $G$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is a complete bipartite graph, Dirac’s Hamiltonicity condition also implies pancyclicity, i.e., existence of cycles of all the lengths from <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000075_inline17.png\" /> <jats:tex-math> $3$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> up to <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000075_inline18.png\" /> <jats:tex-math> $n$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. In this paper, we show that <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000075_inline19.png\" /> <jats:tex-math> $\\delta (G)\\geq \\tilde \\alpha (G)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> implies that <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000075_inline20.png\" /> <jats:tex-math> $G$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is pancyclic or that <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000075_inline21.png\" /> <jats:tex-math> $G=K_{\\frac{n}{2},\\frac{n}{2}}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, thus extending the result of McDiarmid and Yolov, and generalizing the classic theorem of Bondy.","PeriodicalId":10503,"journal":{"name":"Combinatorics, Probability and Computing","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A generalization of Bondy’s pancyclicity theorem\",\"authors\":\"Nemanja Draganić, David Munhá Correia, Benny Sudakov\",\"doi\":\"10.1017/s0963548324000075\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"The <jats:italic>bipartite independence number</jats:italic> of a graph <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S0963548324000075_inline1.png\\\" /> <jats:tex-math> $G$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, denoted as <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S0963548324000075_inline2.png\\\" /> <jats:tex-math> $\\\\tilde \\\\alpha (G)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, is the minimal number <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S0963548324000075_inline3.png\\\" /> <jats:tex-math> $k$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> such that there exist positive integers <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S0963548324000075_inline4.png\\\" /> <jats:tex-math> $a$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S0963548324000075_inline5.png\\\" /> <jats:tex-math> $b$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> with <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S0963548324000075_inline6.png\\\" /> <jats:tex-math> $a+b=k+1$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> with the property that for any two disjoint sets <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S0963548324000075_inline7.png\\\" /> <jats:tex-math> $A,B\\\\subseteq V(G)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> with <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S0963548324000075_inline8.png\\\" /> <jats:tex-math> $|A|=a$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S0963548324000075_inline9.png\\\" /> <jats:tex-math> $|B|=b$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, there is an edge between <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S0963548324000075_inline10.png\\\" /> <jats:tex-math> $A$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S0963548324000075_inline11.png\\\" /> <jats:tex-math> $B$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. McDiarmid and Yolov showed that if <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S0963548324000075_inline12.png\\\" /> <jats:tex-math> $\\\\delta (G)\\\\geq \\\\tilde \\\\alpha (G)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> then <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S0963548324000075_inline13.png\\\" /> <jats:tex-math> $G$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is Hamiltonian, extending the famous theorem of Dirac which states that if <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S0963548324000075_inline14.png\\\" /> <jats:tex-math> $\\\\delta (G)\\\\geq |G|/2$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> then <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S0963548324000075_inline15.png\\\" /> <jats:tex-math> $G$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is Hamiltonian. In 1973, Bondy showed that, unless <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S0963548324000075_inline16.png\\\" /> <jats:tex-math> $G$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is a complete bipartite graph, Dirac’s Hamiltonicity condition also implies pancyclicity, i.e., existence of cycles of all the lengths from <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S0963548324000075_inline17.png\\\" /> <jats:tex-math> $3$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> up to <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S0963548324000075_inline18.png\\\" /> <jats:tex-math> $n$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. In this paper, we show that <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S0963548324000075_inline19.png\\\" /> <jats:tex-math> $\\\\delta (G)\\\\geq \\\\tilde \\\\alpha (G)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> implies that <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S0963548324000075_inline20.png\\\" /> <jats:tex-math> $G$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is pancyclic or that <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" mime-subtype=\\\"png\\\" xlink:href=\\\"S0963548324000075_inline21.png\\\" /> <jats:tex-math> $G=K_{\\\\frac{n}{2},\\\\frac{n}{2}}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, thus extending the result of McDiarmid and Yolov, and generalizing the classic theorem of Bondy.\",\"PeriodicalId\":10503,\"journal\":{\"name\":\"Combinatorics, Probability and Computing\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-04-17\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Combinatorics, Probability and Computing\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1017/s0963548324000075\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Combinatorics, Probability and Computing","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1017/s0963548324000075","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0

摘要

图 $G$ 的双侧独立数表示为 $\tilde \alpha (G)$,是最小数 $k$,使得存在正整数 $a$ 和 $b$,且 $a+b=k+1$ 具有这样的性质:对于任意两个不相交的集合 $A,Bs\subseteq V(G)$,且 $|A|=a$ 和 $|B|=b$ ,在 $A$ 和 $B$ 之间有一条边。McDiarmid 和 Yolov 证明,如果 $\delta (G)\geq \tilde \alpha (G)$ 那么 $G$ 是哈密顿的,这扩展了著名的狄拉克定理,即如果 $\delta (G)\geq |G|/2$ 那么 $G$ 是哈密顿的。1973 年,邦迪证明,除非 $G$ 是一个完整的二叉图,否则狄拉克的哈密顿性条件也意味着泛周期性,即存在从 $3$ 到 $n$ 的所有长度的周期。在本文中,我们证明了 $\delta (G)\geq \tilde \alpha (G)$ 意味着 $G$ 是泛周期的,或者说 $G=K_\{frac{n}{2},\frac{n}{2}$ ,从而扩展了麦克迪尔米德和约洛夫的结果,并推广了邦迪的经典定理。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
A generalization of Bondy’s pancyclicity theorem
The bipartite independence number of a graph $G$ , denoted as $\tilde \alpha (G)$ , is the minimal number $k$ such that there exist positive integers $a$ and $b$ with $a+b=k+1$ with the property that for any two disjoint sets $A,B\subseteq V(G)$ with $|A|=a$ and $|B|=b$ , there is an edge between $A$ and $B$ . McDiarmid and Yolov showed that if $\delta (G)\geq \tilde \alpha (G)$ then $G$ is Hamiltonian, extending the famous theorem of Dirac which states that if $\delta (G)\geq |G|/2$ then $G$ is Hamiltonian. In 1973, Bondy showed that, unless $G$ is a complete bipartite graph, Dirac’s Hamiltonicity condition also implies pancyclicity, i.e., existence of cycles of all the lengths from $3$ up to $n$ . In this paper, we show that $\delta (G)\geq \tilde \alpha (G)$ implies that $G$ is pancyclic or that $G=K_{\frac{n}{2},\frac{n}{2}}$ , thus extending the result of McDiarmid and Yolov, and generalizing the classic theorem of Bondy.
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