{"title":"Constructing flag-transitive, point-primitive 2-designs from complete graphs","authors":"Chuyi Zhong, Shenglin Zhou","doi":"10.1016/j.disc.2024.114217","DOIUrl":null,"url":null,"abstract":"<div><p>In this paper, we study 2-designs <span><math><mi>D</mi><mo>=</mo><mo>(</mo><mi>P</mi><mo>,</mo><msup><mrow><mi>B</mi></mrow><mrow><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></msup><mo>)</mo></math></span>, where <span><math><mi>P</mi></math></span> can be viewed as the edge set of the complete graph <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, and <em>B</em> can be identified as the edge set of a subgraph of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. We give a necessary condition for <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> to be flag-transitive, and then present three ways to construct such 2-designs admitting a flag-transitive, point-primitive automorphism group <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. As an application, all pairs <span><math><mo>(</mo><mi>D</mi><mo>,</mo><mi>G</mi><mo>)</mo></math></span> are determined, where <span><math><mi>D</mi></math></span> is a 2-<span><math><mo>(</mo><mi>v</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>λ</mi><mo>)</mo></math></span> design with <span><math><mi>gcd</mi><mo></mo><mo>(</mo><mi>v</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>=</mo><mn>3</mn></math></span> or 4, and <em>G</em> is flag-transitive with <span><math><mi>S</mi><mi>o</mi><mi>c</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> for <span><math><mi>n</mi><mo>≥</mo><mn>5</mn></math></span>. Furthermore, we show that there are infinite flag-transitive, point-primitive 2-<span><math><mo>(</mo><mi>v</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>λ</mi><mo>)</mo></math></span> designs with <span><math><mi>gcd</mi><mo></mo><mo>(</mo><mi>v</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>≤</mo><msup><mrow><mo>(</mo><mi>v</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></math></span> and alternating socle <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> with <span><math><mi>v</mi><mo>=</mo><mrow><mo>(</mo><mtable><mtr><mtd><mi>n</mi></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr></mtable><mo>)</mo></mrow></math></span>.</p></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 1","pages":"Article 114217"},"PeriodicalIF":0.7000,"publicationDate":"2024-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0012365X24003480/pdfft?md5=8dd45f6c8de1e9aaa5ee26ce47fc990b&pid=1-s2.0-S0012365X24003480-main.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Discrete Mathematics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0012365X24003480","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
Abstract
In this paper, we study 2-designs , where can be viewed as the edge set of the complete graph , and B can be identified as the edge set of a subgraph of . We give a necessary condition for to be flag-transitive, and then present three ways to construct such 2-designs admitting a flag-transitive, point-primitive automorphism group . As an application, all pairs are determined, where is a 2- design with or 4, and G is flag-transitive with for . Furthermore, we show that there are infinite flag-transitive, point-primitive 2- designs with and alternating socle with .
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.