Ann Clifton , Éva Czabarka , Audace A.V. Dossou-Olory , Kevin Liu , Sarah Loeb , Utku Okur , László Székely , Kristina Wicke
{"title":"Decks of rooted binary trees","authors":"Ann Clifton , Éva Czabarka , Audace A.V. Dossou-Olory , Kevin Liu , Sarah Loeb , Utku Okur , László Székely , Kristina Wicke","doi":"10.1016/j.ejc.2024.104076","DOIUrl":"10.1016/j.ejc.2024.104076","url":null,"abstract":"<div><div>We consider extremal problems related to decks and multidecks of rooted binary trees (a.k.a. rooted phylogenetic tree shapes). Here, the deck (resp. multideck) of a tree <span><math><mi>T</mi></math></span> refers to the set (resp. multiset) of leaf-induced binary subtrees of <span><math><mi>T</mi></math></span>. On the one hand, we consider the reconstruction of trees from their (multi)decks. We give lower and upper bounds on the minimum (multi)deck size required to uniquely encode a rooted binary tree on <span><math><mi>n</mi></math></span> leaves. On the other hand, we consider problems related to deck cardinalities. In particular, we characterize trees with minimum-size as well as maximum-size decks. Finally, we present some exhaustive computations for <span><math><mi>k</mi></math></span>-universal trees, i.e., rooted binary trees that contain all <span><math><mi>k</mi></math></span>-leaf rooted binary trees as leaf-induced subtrees.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142318960","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Induced subgraphs and tree decompositions XIV. Non-adjacent neighbours in a hole","authors":"Maria Chudnovsky , Sepehr Hajebi , Sophie Spirkl","doi":"10.1016/j.ejc.2024.104074","DOIUrl":"10.1016/j.ejc.2024.104074","url":null,"abstract":"<div><div>A <em>clock</em> is a graph consisting of an induced cycle <span><math><mi>C</mi></math></span> and a vertex not in <span><math><mi>C</mi></math></span> with at least two non-adjacent neighbours in <span><math><mi>C</mi></math></span>. We show that every clock-free graph of large treewidth contains a “basic obstruction” of large treewidth as an induced subgraph: a complete graph, a subdivision of a wall, or the line graph of a subdivision of a wall.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0195669824001598/pdfft?md5=fa98c8a13265d848775c4b52beb995da&pid=1-s2.0-S0195669824001598-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142315136","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Zig-zag Eulerian polynomials","authors":"T. Kyle Petersen , Yan Zhuang","doi":"10.1016/j.ejc.2024.104073","DOIUrl":"10.1016/j.ejc.2024.104073","url":null,"abstract":"<div><div>For any finite partially ordered set <span><math><mi>P</mi></math></span>, the <span><math><mi>P</mi></math></span>-Eulerian polynomial is the generating function for the descent number over the set of linear extensions of <span><math><mi>P</mi></math></span>, and is closely related to the order polynomial of <span><math><mi>P</mi></math></span> arising in the theory of <span><math><mi>P</mi></math></span>-partitions. Here we study the <span><math><mi>P</mi></math></span>-Eulerian polynomial where <span><math><mi>P</mi></math></span> is a naturally labeled zig-zag poset; we call these <em>zig-zag Eulerian polynomials</em>. A result of Brändén implies that these polynomials are gamma-nonnegative, and hence their coefficients are symmetric and unimodal. The zig-zag Eulerian polynomials and the associated order polynomials have appeared fleetingly in the literature in a wide variety of contexts—e.g., in the study of polytopes, magic labelings of graphs, and Kekulé structures—but they do not appear to have been studied systematically.</div><div>In this paper, we use a “relaxed” version of <span><math><mi>P</mi></math></span>-partitions to both survey and unify results. Our technique shows that the zig-zag Eulerian polynomials also capture the distribution of “big returns” over the set of (up-down) alternating permutations, as first observed by Coons and Sullivant. We develop recurrences for refined versions of the relevant generating functions, which evoke similarities to recurrences for the classical Eulerian polynomials. We conclude with a literature survey and open questions.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0195669824001586/pdfft?md5=899109a09f3bf833016e01aae31a7980&pid=1-s2.0-S0195669824001586-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142315032","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On non-degenerate Turán problems for expansions","authors":"Dániel Gerbner","doi":"10.1016/j.ejc.2024.104071","DOIUrl":"10.1016/j.ejc.2024.104071","url":null,"abstract":"<div><div>The <span><math><mi>r</mi></math></span>-uniform expansion <span><math><msup><mrow><mi>F</mi></mrow><mrow><mrow><mo>(</mo><mi>r</mi><mo>)</mo></mrow><mo>+</mo></mrow></msup></math></span> of a graph <span><math><mi>F</mi></math></span> is obtained by enlarging each edge with <span><math><mrow><mi>r</mi><mo>−</mo><mn>2</mn></mrow></math></span> new vertices such that altogether we use <span><math><mrow><mrow><mo>(</mo><mi>r</mi><mo>−</mo><mn>2</mn><mo>)</mo></mrow><mrow><mo>|</mo><mi>E</mi><mrow><mo>(</mo><mi>F</mi><mo>)</mo></mrow><mo>|</mo></mrow></mrow></math></span> new vertices. Two simple lower bounds on the largest number <span><math><mrow><msub><mrow><mi>ex</mi></mrow><mrow><mi>r</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>,</mo><msup><mrow><mi>F</mi></mrow><mrow><mrow><mo>(</mo><mi>r</mi><mo>)</mo></mrow><mo>+</mo></mrow></msup><mo>)</mo></mrow></mrow></math></span> of <span><math><mi>r</mi></math></span>-edges in <span><math><msup><mrow><mi>F</mi></mrow><mrow><mrow><mo>(</mo><mi>r</mi><mo>)</mo></mrow><mo>+</mo></mrow></msup></math></span>-free <span><math><mi>r</mi></math></span>-graphs are <span><math><mrow><mi>Ω</mi><mrow><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>r</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></mrow></mrow></math></span> (in the case <span><math><mi>F</mi></math></span> is not a star) and <span><math><mrow><mi>ex</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>,</mo><mi>F</mi><mo>)</mo></mrow></mrow></math></span>, which is the largest number of <span><math><mi>r</mi></math></span>-cliques in <span><math><mi>n</mi></math></span>-vertex <span><math><mi>F</mi></math></span>-free graphs. We prove that <span><math><mrow><msub><mrow><mi>ex</mi></mrow><mrow><mi>r</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>,</mo><msup><mrow><mi>F</mi></mrow><mrow><mrow><mo>(</mo><mi>r</mi><mo>)</mo></mrow><mo>+</mo></mrow></msup><mo>)</mo></mrow><mo>=</mo><mi>ex</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>,</mo><mi>F</mi><mo>)</mo></mrow><mo>+</mo><mi>O</mi><mrow><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>r</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></mrow></mrow></math></span>. The proof comes with a structure theorem that we use to determine <span><math><mrow><msub><mrow><mi>ex</mi></mrow><mrow><mi>r</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>,</mo><msup><mrow><mi>F</mi></mrow><mrow><mrow><mo>(</mo><mi>r</mi><mo>)</mo></mrow><mo>+</mo></mrow></msup><mo>)</mo></mrow></mrow></math></span> exactly for some graphs <span><math><mi>F</mi></math></span>, every <span><math><mrow><mi>r</mi><mo><</mo><mi>χ</mi><mrow><mo>(</mo><mi>F</mi><mo>)</mo></mrow></mrow></math></span> and sufficiently large <span><math><mi>n</mi></math></span>.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0195669824001562/pdfft?md5=86fa8d5991cc3c3ff302bc8fdbd50279&pid=1-s2.0-S0195669824001562-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142311821","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Induced subdivisions with pinned branch vertices","authors":"Sepehr Hajebi","doi":"10.1016/j.ejc.2024.104072","DOIUrl":"10.1016/j.ejc.2024.104072","url":null,"abstract":"<div><div>We prove that for all <span><math><mrow><mi>r</mi><mo>∈</mo><mi>N</mi><mo>∪</mo><mrow><mo>{</mo><mn>0</mn><mo>}</mo></mrow></mrow></math></span> and <span><math><mrow><mi>s</mi><mo>,</mo><mi>t</mi><mo>∈</mo><mi>N</mi></mrow></math></span>, there exists <span><math><mrow><mi>Ω</mi><mo>=</mo><mi>Ω</mi><mrow><mo>(</mo><mi>r</mi><mo>,</mo><mi>s</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow><mo>∈</mo><mi>N</mi></mrow></math></span> with the following property. Let <span><math><mi>G</mi></math></span> be a graph and let <span><math><mi>H</mi></math></span> be a subgraph of <span><math><mi>G</mi></math></span> isomorphic to a <span><math><mrow><mo>(</mo><mo>≤</mo><mi>r</mi><mo>)</mo></mrow></math></span>-subdivision of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>Ω</mi></mrow></msub></math></span>. Then either <span><math><mi>G</mi></math></span> contains <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>t</mi></mrow></msub></math></span> or <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>t</mi></mrow></msub></math></span> as an induced subgraph, or there is an induced subgraph <span><math><mi>J</mi></math></span> of <span><math><mi>G</mi></math></span> isomorphic to a proper <span><math><mrow><mo>(</mo><mo>≤</mo><mi>r</mi><mo>)</mo></mrow></math></span>-subdivision of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>s</mi></mrow></msub></math></span> such that every branch vertex of <span><math><mi>J</mi></math></span> is a branch vertex of <span><math><mi>H</mi></math></span>. This answers in the affirmative a question of Lozin and Razgon. In fact, we show that both the branch vertices and the paths corresponding to the subdivided edges between them can be preserved.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0195669824001574/pdfft?md5=ed4f41801de33ce909fbbc25a22d7d22&pid=1-s2.0-S0195669824001574-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142311820","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Set partitions, tableaux, and subspace profiles under regular diagonal matrices","authors":"Amritanshu Prasad , Samrith Ram","doi":"10.1016/j.ejc.2024.104060","DOIUrl":"10.1016/j.ejc.2024.104060","url":null,"abstract":"<div><p>We introduce a family of univariate polynomials indexed by integer partitions. At prime powers, they count the number of subspaces in a finite vector space that transform under a regular diagonal matrix in a specified manner. This enumeration formula is a combinatorial solution to a problem introduced by Bender, Coley, Robbins and Rumsey. At 1, they count set partitions with specified block sizes. At 0, they count standard tableaux of specified shape. At <span><math><mrow><mo>−</mo><mn>1</mn></mrow></math></span>, they count standard shifted tableaux of a specified shape. These polynomials are generated by a new statistic on set partitions (called the interlacing number) as well as a polynomial statistic on standard tableaux. They allow us to express <span><math><mi>q</mi></math></span>-Stirling numbers of the second kind as sums over standard tableaux and as sums over set partitions.</p><p>For partitions whose parts are at most two, these polynomials are the non-zero entries of the Catalan triangle associated to the <span><math><mi>q</mi></math></span>-Hermite orthogonal polynomial sequence. In particular, when all parts are equal to two, they coincide with the polynomials defined by Touchard that enumerate chord diagrams by the number of crossings.</p></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142229590","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Spin models and distance-regular graphs of q-Racah type","authors":"Kazumasa Nomura , Paul Terwilliger","doi":"10.1016/j.ejc.2024.104069","DOIUrl":"10.1016/j.ejc.2024.104069","url":null,"abstract":"<div><p>Let <span><math><mi>Γ</mi></math></span> denote a distance-regular graph, with vertex set <span><math><mi>X</mi></math></span> and diameter <span><math><mrow><mi>D</mi><mo>≥</mo><mn>3</mn></mrow></math></span>. We assume that <span><math><mi>Γ</mi></math></span> is formally self-dual and <span><math><mi>q</mi></math></span>-Racah type. Let <span><math><mi>A</mi></math></span> denote the adjacency matrix of <span><math><mi>Γ</mi></math></span>. Pick <span><math><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow></math></span>, and let <span><math><mrow><msup><mrow><mi>A</mi></mrow><mrow><mo>∗</mo></mrow></msup><mo>=</mo><msup><mrow><mi>A</mi></mrow><mrow><mo>∗</mo></mrow></msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span> denote the dual adjacency matrix of <span><math><mi>Γ</mi></math></span> with respect to <span><math><mi>x</mi></math></span>. The matrices <span><math><mrow><mi>A</mi><mo>,</mo><msup><mrow><mi>A</mi></mrow><mrow><mo>∗</mo></mrow></msup></mrow></math></span> generate the subconstituent algebra <span><math><mrow><mi>T</mi><mo>=</mo><mi>T</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span>. We assume that for every choice of <span><math><mi>x</mi></math></span> the algebra <span><math><mi>T</mi></math></span> contains a certain central element <span><math><mrow><mi>Z</mi><mo>=</mo><mi>Z</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span> whose significance is illuminated by the following relations: <span><span><span><math><mrow><mi>A</mi><mo>+</mo><mfrac><mrow><mi>q</mi><mi>B</mi><mi>C</mi><mo>−</mo><msup><mrow><mi>q</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mi>C</mi><mi>B</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>−</mo><msup><mrow><mi>q</mi></mrow><mrow><mo>−</mo><mn>2</mn></mrow></msup></mrow></mfrac><mo>=</mo><mi>Z</mi><mo>,</mo><mspace></mspace><mi>B</mi><mo>+</mo><mfrac><mrow><mi>q</mi><mi>C</mi><mi>A</mi><mo>−</mo><msup><mrow><mi>q</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mi>A</mi><mi>C</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>−</mo><msup><mrow><mi>q</mi></mrow><mrow><mo>−</mo><mn>2</mn></mrow></msup></mrow></mfrac><mo>=</mo><mi>Z</mi><mo>,</mo><mspace></mspace><mi>C</mi><mo>+</mo><mfrac><mrow><mi>q</mi><mi>A</mi><mi>B</mi><mo>−</mo><msup><mrow><mi>q</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mi>B</mi><mi>A</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>−</mo><msup><mrow><mi>q</mi></mrow><mrow><mo>−</mo><mn>2</mn></mrow></msup></mrow></mfrac><mo>=</mo><mi>Z</mi><mo>.</mo></mrow></math></span></span></span> The matrices <span><math><mi>A</mi></math></span>, <span><math><mi>B</mi></math></span> satisfy <span><math><mrow><mi>A</mi><mo>=</mo><mrow><mo>(</mo><mi>A</mi><mo>−</mo><mi>ɛ</mi><mi>I</mi><mo>)</mo></mrow><mo>/</mo><mi>α</mi></mrow></math></span> and <span><math><mrow><mi>B</mi><mo>=</mo><mrow><mo>(</mo><msup><mrow><mi>A</mi></mrow","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142173819","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"All 3-transitive groups satisfy the strict-Erdős–Ko–Rado property","authors":"Venkata Raghu Tej Pantangi","doi":"10.1016/j.ejc.2024.104057","DOIUrl":"10.1016/j.ejc.2024.104057","url":null,"abstract":"<div><p>A subset <span><math><mi>S</mi></math></span> of a transitive permutation group <span><math><mrow><mi>G</mi><mo>≤</mo><mi>Sym</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span> is said to be an intersecting set if, for every <span><math><mrow><msub><mrow><mi>g</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>g</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>∈</mo><mi>S</mi></mrow></math></span>, there is an <span><math><mrow><mi>i</mi><mo>∈</mo><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow></mrow></math></span> such that <span><math><mrow><msub><mrow><mi>g</mi></mrow><mrow><mn>1</mn></mrow></msub><mrow><mo>(</mo><mi>i</mi><mo>)</mo></mrow><mo>=</mo><msub><mrow><mi>g</mi></mrow><mrow><mn>2</mn></mrow></msub><mrow><mo>(</mo><mi>i</mi><mo>)</mo></mrow></mrow></math></span>. The stabilizer of a point in <span><math><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow></math></span> and its cosets are intersecting sets of size <span><math><mrow><mrow><mo>|</mo><mi>G</mi><mo>|</mo></mrow><mo>/</mo><mi>n</mi></mrow></math></span>. Such families are referred to as canonical intersecting sets. A result by Meagher, Spiga, and Tiep states that if <span><math><mi>G</mi></math></span> is a 2-transitive group, then <span><math><mrow><mrow><mo>|</mo><mi>G</mi><mo>|</mo></mrow><mo>/</mo><mi>n</mi></mrow></math></span> is the size of an intersecting set of maximum size in <span><math><mi>G</mi></math></span>. In some 2-transitive groups (for instance <span><math><mrow><mi>Sym</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span>, <span><math><mrow><mi>Alt</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span>), every intersecting set of maximum possible size is canonical. A permutation group, in which every intersecting family of maximum possible size is canonical, is said to satisfy the strict-EKR property. In this article, we investigate the structure of intersecting sets in 3-transitive groups. A conjecture by Meagher and Spiga states that all 3-transitive groups satisfy the strict-EKR property. Meagher and Spiga showed that this is true for the 3-transitive group <span><math><mrow><mi>PGL</mi><mrow><mo>(</mo><mn>2</mn><mo>,</mo><mi>q</mi><mo>)</mo></mrow></mrow></math></span>. Using the classification of 3-transitive groups and some results in the literature, the conjecture reduces to showing that the 3-transitive group <span><math><mrow><mi>AGL</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mn>2</mn><mo>)</mo></mrow></mrow></math></span> satisfies the strict-EKR property. We show that <span><math><mrow><mi>AGL</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mn>2</mn><mo>)</mo></mrow></mrow></math></span> satisfies the strict-EKR property and as a consequence, we prove Meagher and Spiga’s conjecture. We also prove a stronger result for <span><math><mrow><mi>AGL</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mn>2</mn><mo>)</mo></mrow></mrow></math></span> by showing that “large” intersecting sets in <span><math><mrow><mi>A","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142168242","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Spanning subdivisions in dense digraphs","authors":"Hyunwoo Lee","doi":"10.1016/j.ejc.2024.104059","DOIUrl":"10.1016/j.ejc.2024.104059","url":null,"abstract":"<div><p>We prove that an <span><math><mi>n</mi></math></span>-vertex digraph <span><math><mi>D</mi></math></span> with minimum semi-degree at least <span><math><mrow><mfenced><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>+</mo><mi>ɛ</mi></mrow></mfenced><mi>n</mi></mrow></math></span> and <span><math><mrow><mi>n</mi><mo>≥</mo><mi>C</mi><mi>m</mi></mrow></math></span> contains a subdivision of all <span><math><mi>m</mi></math></span>-arc digraphs without isolated vertices. Here, <span><math><mi>C</mi></math></span> is a constant only depending on <span><math><mrow><mi>ɛ</mi><mo>.</mo></mrow></math></span> This is the best possible and settles a conjecture raised by Pavez-Signé (2023) in a stronger form.</p></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0195669824001446/pdfft?md5=53af666a1aa86ffe42f097ee615130a5&pid=1-s2.0-S0195669824001446-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142148174","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Graphical regular representations of (2,p)-generated groups","authors":"Binzhou Xia","doi":"10.1016/j.ejc.2024.104058","DOIUrl":"10.1016/j.ejc.2024.104058","url":null,"abstract":"<div><p>For groups <span><math><mi>G</mi></math></span> that can be generated by an involution and an element of odd prime order, this paper gives a sufficient condition for a certain Cayley graph of <span><math><mi>G</mi></math></span> to be a graphical regular representation (GRR), that is, for the Cayley graph to have full automorphism group isomorphic to <span><math><mi>G</mi></math></span>. This condition enables one to show the existence of GRRs of prescribed valency for a large class of groups, and in this paper, <span><math><mi>k</mi></math></span>-valent GRRs of finite nonabelian simple groups with <span><math><mrow><mi>k</mi><mo>≥</mo><mn>5</mn></mrow></math></span> are considered.</p></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0195669824001434/pdfft?md5=c7da3e756f2ea49c07fc86ccf367f717&pid=1-s2.0-S0195669824001434-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142148173","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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