Yair Caro , Balázs Patkós , Zsolt Tuza , Máté Vizer
{"title":"图的边缘映射：Turán类型参数","authors":"Yair Caro , Balázs Patkós , Zsolt Tuza , Máté Vizer","doi":"10.1016/j.ejc.2025.104140","DOIUrl":null,"url":null,"abstract":"<div><div>In this paper, we address problems related to parameters concerning edge mappings of graphs. The quantity <span><math><mrow><mi>h</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> is defined to be the maximum number of edges in an <span><math><mi>n</mi></math></span>-vertex graph <span><math><mi>H</mi></math></span> such that there exists a mapping <span><math><mrow><mi>f</mi><mo>:</mo><mi>E</mi><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow><mo>→</mo><mi>E</mi><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow></mrow></math></span> with <span><math><mrow><mi>f</mi><mrow><mo>(</mo><mi>e</mi><mo>)</mo></mrow><mo>≠</mo><mi>e</mi></mrow></math></span> for all <span><math><mrow><mi>e</mi><mo>∈</mo><mi>E</mi><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow></mrow></math></span> and further in all copies <span><math><msup><mrow><mi>G</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span> of <span><math><mi>G</mi></math></span> in <span><math><mi>H</mi></math></span> there exists <span><math><mrow><mi>e</mi><mo>∈</mo><mi>E</mi><mrow><mo>(</mo><msup><mrow><mi>G</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>)</mo></mrow></mrow></math></span> with <span><math><mrow><mi>f</mi><mrow><mo>(</mo><mi>e</mi><mo>)</mo></mrow><mo>∈</mo><mi>E</mi><mrow><mo>(</mo><msup><mrow><mi>G</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>)</mo></mrow></mrow></math></span>. Among other results, we determine <span><math><mrow><mi>h</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> when <span><math><mi>G</mi></math></span> is a matching and <span><math><mi>n</mi></math></span> is large enough.</div><div>As a related concept, we say that <span><math><mi>H</mi></math></span> is unavoidable for <span><math><mi>G</mi></math></span> if for any mapping <span><math><mrow><mi>f</mi><mo>:</mo><mi>E</mi><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow><mo>→</mo><mi>E</mi><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow></mrow></math></span> with <span><math><mrow><mi>f</mi><mrow><mo>(</mo><mi>e</mi><mo>)</mo></mrow><mo>≠</mo><mi>e</mi></mrow></math></span> there exists a copy <span><math><msup><mrow><mi>G</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span> of <span><math><mi>G</mi></math></span> in <span><math><mi>H</mi></math></span> such that <span><math><mrow><mi>f</mi><mrow><mo>(</mo><mi>e</mi><mo>)</mo></mrow><mo>∉</mo><mi>E</mi><mrow><mo>(</mo><msup><mrow><mi>G</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>)</mo></mrow></mrow></math></span> for all <span><math><mrow><mi>e</mi><mo>∈</mo><mi>E</mi><mrow><mo>(</mo><msup><mrow><mi>G</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>)</mo></mrow></mrow></math></span>. The set of minimal unavoidable graphs for <span><math><mi>G</mi></math></span> is denoted by <span><math><mrow><mi>M</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>. We prove that if <span><math><mi>F</mi></math></span> is a forest, then <span><math><mrow><mi>M</mi><mrow><mo>(</mo><mi>F</mi><mo>)</mo></mrow></mrow></math></span> is finite if and only if <span><math><mi>F</mi></math></span> is a matching, and we conjecture that for all non-forest graphs <span><math><mi>G</mi></math></span>, the set <span><math><mrow><mi>M</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> is infinite.</div><div>Several other parameters are defined with basic results proved. Lots of open problems remain.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"127 ","pages":"Article 104140"},"PeriodicalIF":1.0000,"publicationDate":"2025-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Edge mappings of graphs: Turán type parameters\",\"authors\":\"Yair Caro , Balázs Patkós , Zsolt Tuza , Máté Vizer\",\"doi\":\"10.1016/j.ejc.2025.104140\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>In this paper, we address problems related to parameters concerning edge mappings of graphs. The quantity <span><math><mrow><mi>h</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> is defined to be the maximum number of edges in an <span><math><mi>n</mi></math></span>-vertex graph <span><math><mi>H</mi></math></span> such that there exists a mapping <span><math><mrow><mi>f</mi><mo>:</mo><mi>E</mi><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow><mo>→</mo><mi>E</mi><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow></mrow></math></span> with <span><math><mrow><mi>f</mi><mrow><mo>(</mo><mi>e</mi><mo>)</mo></mrow><mo>≠</mo><mi>e</mi></mrow></math></span> for all <span><math><mrow><mi>e</mi><mo>∈</mo><mi>E</mi><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow></mrow></math></span> and further in all copies <span><math><msup><mrow><mi>G</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span> of <span><math><mi>G</mi></math></span> in <span><math><mi>H</mi></math></span> there exists <span><math><mrow><mi>e</mi><mo>∈</mo><mi>E</mi><mrow><mo>(</mo><msup><mrow><mi>G</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>)</mo></mrow></mrow></math></span> with <span><math><mrow><mi>f</mi><mrow><mo>(</mo><mi>e</mi><mo>)</mo></mrow><mo>∈</mo><mi>E</mi><mrow><mo>(</mo><msup><mrow><mi>G</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>)</mo></mrow></mrow></math></span>. Among other results, we determine <span><math><mrow><mi>h</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> when <span><math><mi>G</mi></math></span> is a matching and <span><math><mi>n</mi></math></span> is large enough.</div><div>As a related concept, we say that <span><math><mi>H</mi></math></span> is unavoidable for <span><math><mi>G</mi></math></span> if for any mapping <span><math><mrow><mi>f</mi><mo>:</mo><mi>E</mi><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow><mo>→</mo><mi>E</mi><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow></mrow></math></span> with <span><math><mrow><mi>f</mi><mrow><mo>(</mo><mi>e</mi><mo>)</mo></mrow><mo>≠</mo><mi>e</mi></mrow></math></span> there exists a copy <span><math><msup><mrow><mi>G</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span> of <span><math><mi>G</mi></math></span> in <span><math><mi>H</mi></math></span> such that <span><math><mrow><mi>f</mi><mrow><mo>(</mo><mi>e</mi><mo>)</mo></mrow><mo>∉</mo><mi>E</mi><mrow><mo>(</mo><msup><mrow><mi>G</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>)</mo></mrow></mrow></math></span> for all <span><math><mrow><mi>e</mi><mo>∈</mo><mi>E</mi><mrow><mo>(</mo><msup><mrow><mi>G</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>)</mo></mrow></mrow></math></span>. The set of minimal unavoidable graphs for <span><math><mi>G</mi></math></span> is denoted by <span><math><mrow><mi>M</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>. We prove that if <span><math><mi>F</mi></math></span> is a forest, then <span><math><mrow><mi>M</mi><mrow><mo>(</mo><mi>F</mi><mo>)</mo></mrow></mrow></math></span> is finite if and only if <span><math><mi>F</mi></math></span> is a matching, and we conjecture that for all non-forest graphs <span><math><mi>G</mi></math></span>, the set <span><math><mrow><mi>M</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> is infinite.</div><div>Several other parameters are defined with basic results proved. Lots of open problems remain.</div></div>\",\"PeriodicalId\":50490,\"journal\":{\"name\":\"European Journal of Combinatorics\",\"volume\":\"127 \",\"pages\":\"Article 104140\"},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2025-02-27\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"European Journal of Combinatorics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0195669825000228\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"European Journal of Combinatorics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0195669825000228","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
在本文中，我们讨论了与图的边映射有关的参数问题。量h（n，G）定义为n顶点图h中存在一个映射f:E(h)→E(h)且f(E)≠E的最大边数，对于所有E∈E(h)，并且在h中G的所有拷贝G ‘中存在E∈E（G ’）与f(E)∈E（G '）的映射f:E(h)→E(h)。在其他结果中，当G是匹配且n足够大时，我们确定h（n，G）。作为一个相关的概念,我们说H G如果任何映射f是不可避免的:E (H)→E (H)和f (E)≠E存在G”副本等H G f (E)∉E (G) E∈E (G)。G的最小不可避免图集用M(G)表示。证明了如果F是一个森林，那么M(F)是有限的当且仅当F是一个匹配，并且我们推测对于所有非森林图G，集合M(G)是无限的。定义了其他几个参数，并证明了基本结果。许多悬而未决的问题依然存在。
In this paper, we address problems related to parameters concerning edge mappings of graphs. The quantity is defined to be the maximum number of edges in an -vertex graph such that there exists a mapping with for all and further in all copies of in there exists with . Among other results, we determine when is a matching and is large enough.
As a related concept, we say that is unavoidable for if for any mapping with there exists a copy of in such that for all . The set of minimal unavoidable graphs for is denoted by . We prove that if is a forest, then is finite if and only if is a matching, and we conjecture that for all non-forest graphs , the set is infinite.
Several other parameters are defined with basic results proved. Lots of open problems remain.
期刊介绍：
The European Journal of Combinatorics is a high standard, international, bimonthly journal of pure mathematics, specializing in theories arising from combinatorial problems. The journal is primarily open to papers dealing with mathematical structures within combinatorics and/or establishing direct links between combinatorics and other branches of mathematics and the theories of computing. The journal includes full-length research papers on important topics.
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