{"title":"超图中的横向联盟","authors":"","doi":"10.1016/j.disc.2024.114267","DOIUrl":null,"url":null,"abstract":"<div><div>A transversal in a hypergraph <em>H</em> is set of vertices that intersect every edge of <em>H</em>. A transversal coalition in <em>H</em> consists of two disjoint sets of vertices <em>X</em> and <em>Y</em> of <em>H</em>, neither of which is a transversal but whose union <span><math><mi>X</mi><mo>∪</mo><mi>Y</mi></math></span> is a transversal in <em>H</em>. Such sets <em>X</em> and <em>Y</em> are said to form a transversal coalition. A transversal coalition partition in <em>H</em> is a vertex partition <span><math><mi>Ψ</mi><mo>=</mo><mo>{</mo><msub><mrow><mi>V</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>V</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>V</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>}</mo></math></span> such that for all <span><math><mi>i</mi><mo>∈</mo><mo>[</mo><mi>p</mi><mo>]</mo></math></span>, either the set <span><math><msub><mrow><mi>V</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> is a singleton set that is a transversal in <em>H</em> or the set <span><math><msub><mrow><mi>V</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> forms a transversal coalition with another set <span><math><msub><mrow><mi>V</mi></mrow><mrow><mi>j</mi></mrow></msub></math></span> for some <em>j</em>, where <span><math><mi>j</mi><mo>∈</mo><mo>[</mo><mi>p</mi><mo>]</mo><mo>∖</mo><mo>{</mo><mi>i</mi><mo>}</mo></math></span>. The transversal coalition number <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>τ</mi></mrow></msub><mo>(</mo><mi>H</mi><mo>)</mo></math></span> in <em>H</em> equals the maximum order of a transversal coalition partition in <em>H</em>. For <span><math><mi>k</mi><mo>≥</mo><mn>2</mn></math></span> a hypergraph <em>H</em> is <em>k</em>-uniform if every edge of <em>H</em> has cardinality <em>k</em>. Among other results, we prove that if <span><math><mi>k</mi><mo>≥</mo><mn>2</mn></math></span> and <em>H</em> is a <em>k</em>-uniform hypergraph, then <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>τ</mi></mrow></msub><mo>(</mo><mi>H</mi><mo>)</mo><mo>≤</mo><mo>⌊</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac><msup><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>⌋</mo><mo>+</mo><mi>k</mi><mo>+</mo><mn>1</mn></math></span>. Further we show that for every <span><math><mi>k</mi><mo>≥</mo><mn>2</mn></math></span>, there exists a <em>k</em>-uniform hypergraph that achieves equality in this upper bound.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7000,"publicationDate":"2024-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Transversal coalitions in hypergraphs\",\"authors\":\"\",\"doi\":\"10.1016/j.disc.2024.114267\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>A transversal in a hypergraph <em>H</em> is set of vertices that intersect every edge of <em>H</em>. A transversal coalition in <em>H</em> consists of two disjoint sets of vertices <em>X</em> and <em>Y</em> of <em>H</em>, neither of which is a transversal but whose union <span><math><mi>X</mi><mo>∪</mo><mi>Y</mi></math></span> is a transversal in <em>H</em>. Such sets <em>X</em> and <em>Y</em> are said to form a transversal coalition. A transversal coalition partition in <em>H</em> is a vertex partition <span><math><mi>Ψ</mi><mo>=</mo><mo>{</mo><msub><mrow><mi>V</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>V</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>V</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>}</mo></math></span> such that for all <span><math><mi>i</mi><mo>∈</mo><mo>[</mo><mi>p</mi><mo>]</mo></math></span>, either the set <span><math><msub><mrow><mi>V</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> is a singleton set that is a transversal in <em>H</em> or the set <span><math><msub><mrow><mi>V</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> forms a transversal coalition with another set <span><math><msub><mrow><mi>V</mi></mrow><mrow><mi>j</mi></mrow></msub></math></span> for some <em>j</em>, where <span><math><mi>j</mi><mo>∈</mo><mo>[</mo><mi>p</mi><mo>]</mo><mo>∖</mo><mo>{</mo><mi>i</mi><mo>}</mo></math></span>. The transversal coalition number <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>τ</mi></mrow></msub><mo>(</mo><mi>H</mi><mo>)</mo></math></span> in <em>H</em> equals the maximum order of a transversal coalition partition in <em>H</em>. For <span><math><mi>k</mi><mo>≥</mo><mn>2</mn></math></span> a hypergraph <em>H</em> is <em>k</em>-uniform if every edge of <em>H</em> has cardinality <em>k</em>. Among other results, we prove that if <span><math><mi>k</mi><mo>≥</mo><mn>2</mn></math></span> and <em>H</em> is a <em>k</em>-uniform hypergraph, then <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>τ</mi></mrow></msub><mo>(</mo><mi>H</mi><mo>)</mo><mo>≤</mo><mo>⌊</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac><msup><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>⌋</mo><mo>+</mo><mi>k</mi><mo>+</mo><mn>1</mn></math></span>. Further we show that for every <span><math><mi>k</mi><mo>≥</mo><mn>2</mn></math></span>, there exists a <em>k</em>-uniform hypergraph that achieves equality in this upper bound.</div></div>\",\"PeriodicalId\":50572,\"journal\":{\"name\":\"Discrete Mathematics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.7000,\"publicationDate\":\"2024-09-24\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Discrete Mathematics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0012365X24003984\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Discrete Mathematics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0012365X24003984","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
超图 H 中的横向是指与 H 的每条边相交的顶点集合。H 中的横向联盟由 H 的两个不相交的顶点集合 X 和 Y 组成,这两个集合都不是横向,但它们的结合 X∪Y 是 H 中的横向。H 中的横向联盟分区是一个顶点分区Ψ={V1,V2,...,Vp},对于所有 i∈[p],要么集合 Vi 是 H 中横向的单子集,要么集合 Vi 与某个 j 的另一个集合 Vj 形成横向联盟,其中 j∈[p]∖{i}。H 中的横向联盟数 Cτ(H) 等于 H 中横向联盟分区的最大阶数。对于 k≥2 的超图 H,如果 H 中的每条边都有 cardinality k,那么 H 就是 k-uniform 的。我们进一步证明,对于每一个 k≥2,都存在一个达到这个上界相等的 k-uniform 超图。
A transversal in a hypergraph H is set of vertices that intersect every edge of H. A transversal coalition in H consists of two disjoint sets of vertices X and Y of H, neither of which is a transversal but whose union is a transversal in H. Such sets X and Y are said to form a transversal coalition. A transversal coalition partition in H is a vertex partition such that for all , either the set is a singleton set that is a transversal in H or the set forms a transversal coalition with another set for some j, where . The transversal coalition number in H equals the maximum order of a transversal coalition partition in H. For a hypergraph H is k-uniform if every edge of H has cardinality k. Among other results, we prove that if and H is a k-uniform hypergraph, then . Further we show that for every , there exists a k-uniform hypergraph that achieves equality in this upper bound.
期刊介绍:
Discrete Mathematics provides a common forum for significant research in many areas of discrete mathematics and combinatorics. Among the fields covered by Discrete Mathematics are graph and hypergraph theory, enumeration, coding theory, block designs, the combinatorics of partially ordered sets, extremal set theory, matroid theory, algebraic combinatorics, discrete geometry, matrices, and discrete probability theory.
Items in the journal include research articles (Contributions or Notes, depending on length) and survey/expository articles (Perspectives). Efforts are made to process the submission of Notes (short articles) quickly. The Perspectives section features expository articles accessible to a broad audience that cast new light or present unifying points of view on well-known or insufficiently-known topics.