European Journal of Combinatorics最新文献

{"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>&lt;</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":"124 ","pages":"Article 104071"},"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":"124 ","pages":"Article 104072"},"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 ,&nbsp;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":"124 ","pages":"Article 104060"},"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 ,&nbsp;Paul Terwilliger","doi":"10.1016/j.ejc.2024.104069","DOIUrl":"10.1016/j.ejc.2024.104069","url":null,"abstract":"&lt;div&gt;&lt;p&gt;Let &lt;span&gt;&lt;math&gt;&lt;mi&gt;Γ&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; denote a distance-regular graph, with vertex set &lt;span&gt;&lt;math&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; and diameter &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;D&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;. We assume that &lt;span&gt;&lt;math&gt;&lt;mi&gt;Γ&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; is formally self-dual and &lt;span&gt;&lt;math&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;-Racah type. Let &lt;span&gt;&lt;math&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; denote the adjacency matrix of &lt;span&gt;&lt;math&gt;&lt;mi&gt;Γ&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;. Pick &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;, and let &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;∗&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;∗&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; denote the dual adjacency matrix of &lt;span&gt;&lt;math&gt;&lt;mi&gt;Γ&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; with respect to &lt;span&gt;&lt;math&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;. The matrices &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;∗&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; generate the subconstituent algebra &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;. We assume that for every choice of &lt;span&gt;&lt;math&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; the algebra &lt;span&gt;&lt;math&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; contains a certain central element &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; whose significance is illuminated by the following relations: &lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; The matrices &lt;span&gt;&lt;math&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; satisfy &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;ɛ&lt;/mi&gt;&lt;mi&gt;I&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"124 ","pages":"Article 104069"},"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":"&lt;div&gt;&lt;p&gt;A subset &lt;span&gt;&lt;math&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; of a transitive permutation group &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mi&gt;Sym&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; is said to be an intersecting set if, for every &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;g&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;g&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;, there is an &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; such that &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;g&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;g&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;. The stabilizer of a point in &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; and its cosets are intersecting sets of size &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;. Such families are referred to as canonical intersecting sets. A result by Meagher, Spiga, and Tiep states that if &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; is a 2-transitive group, then &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; is the size of an intersecting set of maximum size in &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;. In some 2-transitive groups (for instance &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;Sym&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;Alt&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;), 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 &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;PGL&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;. Using the classification of 3-transitive groups and some results in the literature, the conjecture reduces to showing that the 3-transitive group &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;AGL&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; satisfies the strict-EKR property. We show that &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;AGL&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; satisfies the strict-EKR property and as a consequence, we prove Meagher and Spiga’s conjecture. We also prove a stronger result for &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;AGL&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; by showing that “large” intersecting sets in &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;A","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"124 ","pages":"Article 104057"},"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":"124 ","pages":"Article 104059"},"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":"124 ","pages":"Article 104058"},"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}

{"title":"Quasi-transitive K∞-minor free graphs","authors":"Matthias Hamann","doi":"10.1016/j.ejc.2024.104056","DOIUrl":"10.1016/j.ejc.2024.104056","url":null,"abstract":"<div><p>We prove that every locally finite quasi-transitive graph that does not contain <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>∞</mi></mrow></msub></math></span> as a minor is quasi-isometric to some planar quasi-transitive locally finite graph. This solves a problem of Esperet and Giocanti and improves their recent result that such graphs are quasi-isometric to some planar graph of bounded degree.</p></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"124 ","pages":"Article 104056"},"PeriodicalIF":1.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0195669824001410/pdfft?md5=363aad4468d615e2be63ead62bf9c355&pid=1-s2.0-S0195669824001410-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142148172","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":"Chromatic quasisymmetric class functions for combinatorial Hopf monoids","authors":"Jacob A. White","doi":"10.1016/j.ejc.2024.104055","DOIUrl":"10.1016/j.ejc.2024.104055","url":null,"abstract":"<div><p>We study the chromatic quasisymmetric class function of a linearized combinatorial Hopf monoid. Given a linearized combinatorial Hopf monoid <span><math><mi>H</mi></math></span>, and an <span><math><mi>H</mi></math></span>-structure <span><math><mi>h</mi></math></span> on a set <span><math><mi>N</mi></math></span>, there are proper colorings of <span><math><mi>h</mi></math></span>, generalizing graph colorings and poset partitions. We show that the automorphism group of <span><math><mi>h</mi></math></span> acts on the set of proper colorings. The chromatic quasisymmetric class function enumerates the fixed points of this action, weighting each coloring with a monomial. For the Hopf monoid of graphs this invariant generalizes Stanley’s chromatic symmetric function and specializes to the orbital chromatic polynomial of Cameron and Kayibi. We deduce various inequalities for the associated orbital polynomial invariants. We apply these results to several examples related to enumerating graph colorings, poset partitions, generic functions on matroids or generalized permutohedra, and others.</p></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"124 ","pages":"Article 104055"},"PeriodicalIF":1.0,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0195669824001409/pdfft?md5=63ba3288644e8ff2f9de5b9b878244e1&pid=1-s2.0-S0195669824001409-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142083870","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 balanceable and simply balanceable regular graphs","authors":"Milad Ahanjideh ,&nbsp;Martin Milanič ,&nbsp;Mary Servatius","doi":"10.1016/j.ejc.2024.104045","DOIUrl":"10.1016/j.ejc.2024.104045","url":null,"abstract":"<div><p>We continue the study of balanceable graphs, defined by Caro, Hansberg, and Montejano in 2021 as graphs <span><math><mi>G</mi></math></span> such that any 2-coloring of the edges of a sufficiently large complete graph containing sufficiently many edges of each color contains a balanced copy of <span><math><mi>G</mi></math></span> (that is, a copy with half the edges of each color). While the problem of recognizing balanceable graphs was conjectured to be <span><math><mi>NP</mi></math></span>-complete by Dailly, Hansberg, and Ventura in 2021, balanceable graphs admit an elegant combinatorial characterization: a graph is balanceable if and only there exist two vertex subsets, one containing half of all the graph’s edges and another one such that the corresponding cut contains half of all the graph’s edges. We consider a special case of this property, namely when one of the two sets is a vertex cover, and call the corresponding graphs simply balanceable. We prove a number of results on balanceable and simply balanceable regular graphs. First, we characterize simply balanceable regular graphs via a condition involving the independence number of the graph. Second, we address a question of Dailly, Hansberg, and Ventura from 2021 and show that every cubic graph is balanceable. Third, using Brooks’ theorem, we show that every 4-regular graph with order divisible by 4 is balanceable. Finally, we show that it is <span><math><mi>NP</mi></math></span>-complete to determine if a 9-regular graph is simply balanceable.</p></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"124 ","pages":"Article 104045"},"PeriodicalIF":1.0,"publicationDate":"2024-08-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0195669824001306/pdfft?md5=129ec5810b9eb2e86760507f0c3a96ba&pid=1-s2.0-S0195669824001306-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142048283","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}