Pub Date : 2026-08-01Epub Date: 2026-02-23DOI: 10.1016/j.jcta.2026.106180
Martha Du Preez, William Q. Erickson, Jonathan Feigert, Markus Hunziker, Jonathan Meddaugh, Mitchell Minyard, Kyle Rosengartner, Mark R. Sepanski
In the symmetric group , each element σ has an associated cycle type α, a partition of n that identifies the conjugacy class of σ. The Robinson–Schensted (RS) correspondence links each σ to another partition λ of n, representing the shape of the pair of Young tableaux produced by applying the RS row-insertion algorithm to σ. Surprisingly, the relationship between these two partitions, namely the cycle type α and the RS shape λ, has only recently become a subject of study. In this work, we explicitly describe the set of RS shapes λ that can arise from elements of each cycle type α in cases where α consists of two cycles. To do this, we introduce the notion of an α-coloring, where one colors the entries in a certain tableau of shape λ, in such a way as to construct a permutation σ with cycle type α and RS shape λ.
{"title":"Robinson–Schensted shapes arising from cycle decompositions","authors":"Martha Du Preez, William Q. Erickson, Jonathan Feigert, Markus Hunziker, Jonathan Meddaugh, Mitchell Minyard, Kyle Rosengartner, Mark R. Sepanski","doi":"10.1016/j.jcta.2026.106180","DOIUrl":"10.1016/j.jcta.2026.106180","url":null,"abstract":"<div><div>In the symmetric group <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, each element <em>σ</em> has an associated cycle type <em>α</em>, a partition of <em>n</em> that identifies the conjugacy class of <em>σ</em>. The Robinson–Schensted (RS) correspondence links each <em>σ</em> to another partition <em>λ</em> of <em>n</em>, representing the shape of the pair of Young tableaux produced by applying the RS row-insertion algorithm to <em>σ</em>. Surprisingly, the relationship between these two partitions, namely the cycle type <em>α</em> and the RS shape <em>λ</em>, has only recently become a subject of study. In this work, we explicitly describe the set of RS shapes <em>λ</em> that can arise from elements of each cycle type <em>α</em> in cases where <em>α</em> consists of two cycles. To do this, we introduce the notion of an <em>α-coloring</em>, where one colors the entries in a certain tableau of shape <em>λ</em>, in such a way as to construct a permutation <em>σ</em> with cycle type <em>α</em> and RS shape <em>λ</em>.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"222 ","pages":"Article 106180"},"PeriodicalIF":1.2,"publicationDate":"2026-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146778551","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-08-01Epub Date: 2026-02-03DOI: 10.1016/j.jcta.2026.106164
Mingqing Zhai , Longfei Fang , Huiqiu Lin
<div><div>Minors play a crucial role in various branches of graph theory, including structural graph theory, extremal graph theory, and topological graph theory, and have garnered significant interest in these areas. This paper explores the maximal spectral radius, denoted <span><math><mi>s</mi><mi>p</mi><mi>e</mi><mi>x</mi><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>H</mi></mrow><mrow><mi>m</mi><mi>i</mi><mi>n</mi><mi>o</mi><mi>r</mi></mrow></msub><mo>)</mo></math></span>, of <em>n</em>-vertex graphs that exclude any graph from a fixed family <span><math><mi>H</mi></math></span> as a minor.</div><div>We derive the asymptotic value for <span><math><mi>s</mi><mi>p</mi><mi>e</mi><mi>x</mi><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>H</mi></mrow><mrow><mi>m</mi><mi>i</mi><mi>n</mi><mi>o</mi><mi>r</mi></mrow></msub><mo>)</mo></math></span> and establish a unified stable structure for extremal graphs by introducing a novel application of the absorbing method to eigenvalue analysis, along with models and partitions with respect to a general <em>H</em> minor. In particular, we prove three central theorems, the most fundamental of which asserts that every graph with spectral radius <span><math><mi>ρ</mi><mo>≥</mo><mi>s</mi><mi>p</mi><mi>e</mi><mi>x</mi><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mo>{</mo><mi>H</mi><mo>}</mo></mrow><mrow><mi>m</mi><mi>i</mi><mi>n</mi><mi>o</mi><mi>r</mi></mrow></msub><mo>)</mo></math></span> contains either an <em>H</em> minor or a spanning book <span><math><msub><mrow><mi>B</mi></mrow><mrow><msub><mrow><mi>γ</mi></mrow><mrow><mi>H</mi></mrow></msub><mo>,</mo><mi>n</mi><mo>−</mo><msub><mrow><mi>γ</mi></mrow><mrow><mi>H</mi></mrow></msub></mrow></msub></math></span>, where <span><math><msub><mrow><mi>γ</mi></mrow><mrow><mi>H</mi></mrow></msub><mo>=</mo><mo>|</mo><mi>H</mi><mo>|</mo><mo>−</mo><msub><mrow><mi>α</mi></mrow><mrow><mi>H</mi></mrow></msub><mo>−</mo><mn>1</mn></math></span> and <span><math><msub><mrow><mi>α</mi></mrow><mrow><mi>H</mi></mrow></msub></math></span> is the independence number of <em>H</em>.</div><div>These three theorems, combined with detailed combinatorial analysis, enable us to determine <span><math><mi>s</mi><mi>p</mi><mi>e</mi><mi>x</mi><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mo>{</mo><mi>H</mi><mo>}</mo></mrow><mrow><mi>m</mi><mi>i</mi><mi>n</mi><mi>o</mi><mi>r</mi></mrow></msub><mo>)</mo></math></span> for every complete <em>r</em>-partite graph <em>H</em>. This extends the result of Tait for <span><math><mi>s</mi><mi>p</mi><mi>e</mi><mi>x</mi><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mo>{</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>}</mo></mrow><mrow><mi>m</mi><mi>i</mi><mi>n</mi><mi>o</mi><mi>r</mi></mrow></msub><mo>)</mo></math></span> and provides a stronger solution to his conjecture for <span><math><mi>s</mi><mi>p</mi><mi>e</mi><mi>x</mi><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mo>{</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mro
{"title":"Extremal eigenvalues with respect to graph minors","authors":"Mingqing Zhai , Longfei Fang , Huiqiu Lin","doi":"10.1016/j.jcta.2026.106164","DOIUrl":"10.1016/j.jcta.2026.106164","url":null,"abstract":"<div><div>Minors play a crucial role in various branches of graph theory, including structural graph theory, extremal graph theory, and topological graph theory, and have garnered significant interest in these areas. This paper explores the maximal spectral radius, denoted <span><math><mi>s</mi><mi>p</mi><mi>e</mi><mi>x</mi><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>H</mi></mrow><mrow><mi>m</mi><mi>i</mi><mi>n</mi><mi>o</mi><mi>r</mi></mrow></msub><mo>)</mo></math></span>, of <em>n</em>-vertex graphs that exclude any graph from a fixed family <span><math><mi>H</mi></math></span> as a minor.</div><div>We derive the asymptotic value for <span><math><mi>s</mi><mi>p</mi><mi>e</mi><mi>x</mi><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>H</mi></mrow><mrow><mi>m</mi><mi>i</mi><mi>n</mi><mi>o</mi><mi>r</mi></mrow></msub><mo>)</mo></math></span> and establish a unified stable structure for extremal graphs by introducing a novel application of the absorbing method to eigenvalue analysis, along with models and partitions with respect to a general <em>H</em> minor. In particular, we prove three central theorems, the most fundamental of which asserts that every graph with spectral radius <span><math><mi>ρ</mi><mo>≥</mo><mi>s</mi><mi>p</mi><mi>e</mi><mi>x</mi><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mo>{</mo><mi>H</mi><mo>}</mo></mrow><mrow><mi>m</mi><mi>i</mi><mi>n</mi><mi>o</mi><mi>r</mi></mrow></msub><mo>)</mo></math></span> contains either an <em>H</em> minor or a spanning book <span><math><msub><mrow><mi>B</mi></mrow><mrow><msub><mrow><mi>γ</mi></mrow><mrow><mi>H</mi></mrow></msub><mo>,</mo><mi>n</mi><mo>−</mo><msub><mrow><mi>γ</mi></mrow><mrow><mi>H</mi></mrow></msub></mrow></msub></math></span>, where <span><math><msub><mrow><mi>γ</mi></mrow><mrow><mi>H</mi></mrow></msub><mo>=</mo><mo>|</mo><mi>H</mi><mo>|</mo><mo>−</mo><msub><mrow><mi>α</mi></mrow><mrow><mi>H</mi></mrow></msub><mo>−</mo><mn>1</mn></math></span> and <span><math><msub><mrow><mi>α</mi></mrow><mrow><mi>H</mi></mrow></msub></math></span> is the independence number of <em>H</em>.</div><div>These three theorems, combined with detailed combinatorial analysis, enable us to determine <span><math><mi>s</mi><mi>p</mi><mi>e</mi><mi>x</mi><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mo>{</mo><mi>H</mi><mo>}</mo></mrow><mrow><mi>m</mi><mi>i</mi><mi>n</mi><mi>o</mi><mi>r</mi></mrow></msub><mo>)</mo></math></span> for every complete <em>r</em>-partite graph <em>H</em>. This extends the result of Tait for <span><math><mi>s</mi><mi>p</mi><mi>e</mi><mi>x</mi><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mo>{</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>}</mo></mrow><mrow><mi>m</mi><mi>i</mi><mi>n</mi><mi>o</mi><mi>r</mi></mrow></msub><mo>)</mo></math></span> and provides a stronger solution to his conjecture for <span><math><mi>s</mi><mi>p</mi><mi>e</mi><mi>x</mi><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mo>{</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mro","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"222 ","pages":"Article 106164"},"PeriodicalIF":1.2,"publicationDate":"2026-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146098576","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-08-01Epub Date: 2026-02-23DOI: 10.1016/j.jcta.2026.106177
Zhilin Zhang , Delu Tian , Shenglin Zhou
In this paper, we determine the quadruple systems admitting an almost simple group with socle as the block-transitive and point-2-transitive automorphism group. As a consequence, we classify all flag-transitive quadruple systems by using the classification of finite 2-transitive permutation groups.
{"title":"On quadruple systems with index λ","authors":"Zhilin Zhang , Delu Tian , Shenglin Zhou","doi":"10.1016/j.jcta.2026.106177","DOIUrl":"10.1016/j.jcta.2026.106177","url":null,"abstract":"<div><div>In this paper, we determine the quadruple systems admitting an almost simple group with socle <span><math><mi>P</mi><mi>S</mi><msub><mrow><mi>L</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>q</mi><mo>)</mo></math></span> as the block-transitive and point-2-transitive automorphism group. As a consequence, we classify all flag-transitive quadruple systems by using the classification of finite 2-transitive permutation groups.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"222 ","pages":"Article 106177"},"PeriodicalIF":1.2,"publicationDate":"2026-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147278422","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-08-01Epub Date: 2026-03-06DOI: 10.1016/j.jcta.2026.106181
Haihua Deng , Hexiang Huang , Qing Xiang
<div><div>Griesmer codes are linear codes meeting the Griesmer bound. A linear code is called Δ-divisible if the weights of all codewords are divisible by Δ. In this paper, we investigate the divisibility of Griesmer codes. Let <em>C</em> be an <span><math><msub><mrow><mo>[</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>d</mi><mo>]</mo></mrow><mrow><mi>q</mi></mrow></msub></math></span> Griesmer code with <span><math><mi>q</mi><mo>=</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>f</mi></mrow></msup></math></span>, where <em>p</em> is a prime and <span><math><mi>f</mi><mo>≥</mo><mn>1</mn></math></span> is an integer. In earlier work, Ward proved: (1) for prime fields (i.e., <span><math><mi>q</mi><mo>=</mo><mi>p</mi></math></span>), if <span><math><msup><mrow><mi>p</mi></mrow><mrow><mi>e</mi></mrow></msup><mo>|</mo><mi>d</mi></math></span>, then <em>C</em> is <span><math><msup><mrow><mi>p</mi></mrow><mrow><mi>e</mi></mrow></msup></math></span>-divisible; (2) if <span><math><mi>q</mi><mo>|</mo><mi>d</mi></math></span>, then <em>C</em> is <em>p</em>-divisible; (3) when <span><math><mi>q</mi><mo>=</mo><mn>4</mn></math></span>, if <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>e</mi></mrow></msup><mo>|</mo><mi>d</mi></math></span>, then <em>C</em> is <span><math><mo>⌈</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>e</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>⌉</mo></math></span>-divisible. Based on these results, Ward conjectured that if <span><math><msup><mrow><mi>p</mi></mrow><mrow><mi>e</mi></mrow></msup><mo>|</mo><mi>d</mi></math></span>, then <em>C</em> is <span><math><mo>⌈</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>e</mi><mo>−</mo><mo>(</mo><mi>f</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></msup><mo>⌉</mo></math></span>-divisible.</div><div>In the present work, we obtain two new results on divisibility of a Griesmer <span><math><msub><mrow><mo>[</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>d</mi><mo>]</mo></mrow><mrow><mi>q</mi></mrow></msub></math></span> code <em>C</em>: (a) if <span><math><msup><mrow><mi>q</mi></mrow><mrow><mi>e</mi></mrow></msup><mo>|</mo><mi>d</mi></math></span>, then <em>C</em> is <span><math><msup><mrow><mi>p</mi></mrow><mrow><mi>e</mi></mrow></msup></math></span>-divisible; (b) if <span><math><msup><mrow><mi>p</mi></mrow><mrow><mi>e</mi></mrow></msup><mo>|</mo><mi>d</mi></math></span>, then <em>C</em> is <span><math><mo>⌈</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>e</mi><mo>−</mo><mo>(</mo><mi>f</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>(</mo><mi>q</mi><mo>−</mo><mn>2</mn><mo>)</mo></mrow></msup><mo>⌉</mo></math></span>-divisible. To prove these results, we first show that any <span><math><msub><mrow><mo>[</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>d</mi><mo>]</mo></mrow><mrow><mi>q</mi></mrow></msub></math></span> Griesmer code <em>C</em> admits an ordered basis consisting of <em>k</em> codewords such that the first <em>i</em> of them span a Griesmer subcode for any <span><math><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>k
{"title":"Divisibility of Griesmer codes","authors":"Haihua Deng , Hexiang Huang , Qing Xiang","doi":"10.1016/j.jcta.2026.106181","DOIUrl":"10.1016/j.jcta.2026.106181","url":null,"abstract":"<div><div>Griesmer codes are linear codes meeting the Griesmer bound. A linear code is called Δ-divisible if the weights of all codewords are divisible by Δ. In this paper, we investigate the divisibility of Griesmer codes. Let <em>C</em> be an <span><math><msub><mrow><mo>[</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>d</mi><mo>]</mo></mrow><mrow><mi>q</mi></mrow></msub></math></span> Griesmer code with <span><math><mi>q</mi><mo>=</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>f</mi></mrow></msup></math></span>, where <em>p</em> is a prime and <span><math><mi>f</mi><mo>≥</mo><mn>1</mn></math></span> is an integer. In earlier work, Ward proved: (1) for prime fields (i.e., <span><math><mi>q</mi><mo>=</mo><mi>p</mi></math></span>), if <span><math><msup><mrow><mi>p</mi></mrow><mrow><mi>e</mi></mrow></msup><mo>|</mo><mi>d</mi></math></span>, then <em>C</em> is <span><math><msup><mrow><mi>p</mi></mrow><mrow><mi>e</mi></mrow></msup></math></span>-divisible; (2) if <span><math><mi>q</mi><mo>|</mo><mi>d</mi></math></span>, then <em>C</em> is <em>p</em>-divisible; (3) when <span><math><mi>q</mi><mo>=</mo><mn>4</mn></math></span>, if <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>e</mi></mrow></msup><mo>|</mo><mi>d</mi></math></span>, then <em>C</em> is <span><math><mo>⌈</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>e</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>⌉</mo></math></span>-divisible. Based on these results, Ward conjectured that if <span><math><msup><mrow><mi>p</mi></mrow><mrow><mi>e</mi></mrow></msup><mo>|</mo><mi>d</mi></math></span>, then <em>C</em> is <span><math><mo>⌈</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>e</mi><mo>−</mo><mo>(</mo><mi>f</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></msup><mo>⌉</mo></math></span>-divisible.</div><div>In the present work, we obtain two new results on divisibility of a Griesmer <span><math><msub><mrow><mo>[</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>d</mi><mo>]</mo></mrow><mrow><mi>q</mi></mrow></msub></math></span> code <em>C</em>: (a) if <span><math><msup><mrow><mi>q</mi></mrow><mrow><mi>e</mi></mrow></msup><mo>|</mo><mi>d</mi></math></span>, then <em>C</em> is <span><math><msup><mrow><mi>p</mi></mrow><mrow><mi>e</mi></mrow></msup></math></span>-divisible; (b) if <span><math><msup><mrow><mi>p</mi></mrow><mrow><mi>e</mi></mrow></msup><mo>|</mo><mi>d</mi></math></span>, then <em>C</em> is <span><math><mo>⌈</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>e</mi><mo>−</mo><mo>(</mo><mi>f</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>(</mo><mi>q</mi><mo>−</mo><mn>2</mn><mo>)</mo></mrow></msup><mo>⌉</mo></math></span>-divisible. To prove these results, we first show that any <span><math><msub><mrow><mo>[</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>d</mi><mo>]</mo></mrow><mrow><mi>q</mi></mrow></msub></math></span> Griesmer code <em>C</em> admits an ordered basis consisting of <em>k</em> codewords such that the first <em>i</em> of them span a Griesmer subcode for any <span><math><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>k","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"222 ","pages":"Article 106181"},"PeriodicalIF":1.2,"publicationDate":"2026-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147385838","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-07-01Epub Date: 2026-01-29DOI: 10.1016/j.jcta.2026.106167
Primož Šparl , Jin-Xin Zhou
A graph is said to be half-arc-transitive if its automorphism group is transitive on the vertices and the edges of the graph but not on its arcs. Tetravalent half-arc-transitive graphs of girth 3 were characterized by Marušič and Xu in 1997, while those of girth 4 were characterized by Marušič and Nedela in 2002 and by Potočnik and Wilson in 2007. The investigation of tetravalent half-arc-transitive graphs of girth 5 was initiated by Antončič and Šparl in 2023. In this paper, a characterization of all tetravalent half-arc-transitive graphs of girth 5 is given, and as an application, two open questions from Antončič and Šparl (2023) [1] are answered.
{"title":"A characterization of tetravalent half-arc-transitive graphs of girth 5","authors":"Primož Šparl , Jin-Xin Zhou","doi":"10.1016/j.jcta.2026.106167","DOIUrl":"10.1016/j.jcta.2026.106167","url":null,"abstract":"<div><div>A graph is said to be <em>half-arc-transitive</em> if its automorphism group is transitive on the vertices and the edges of the graph but not on its arcs. Tetravalent half-arc-transitive graphs of girth 3 were characterized by Marušič and Xu in 1997, while those of girth 4 were characterized by Marušič and Nedela in 2002 and by Potočnik and Wilson in 2007. The investigation of tetravalent half-arc-transitive graphs of girth 5 was initiated by Antončič and Šparl in 2023. In this paper, a characterization of all tetravalent half-arc-transitive graphs of girth 5 is given, and as an application, two open questions from Antončič and Šparl (2023) <span><span>[1]</span></span> are answered.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"221 ","pages":"Article 106167"},"PeriodicalIF":1.2,"publicationDate":"2026-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146071596","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-07-01Epub Date: 2026-01-26DOI: 10.1016/j.jcta.2026.106163
Hao Huang , Yi Zhang
A family of subsets is intersecting if for any . In this paper, we show that for given integers and , and any intersecting family of k-subsets of , there exists a d-subset of contained in at most subsets of . This result, proved using spectral graph theory, gives a d-degree generalization of the celebrated Erdős–Ko–Rado Theorem, improving a theorem of Kupavskii.
{"title":"On a d-degree Erdős–Ko–Rado Theorem","authors":"Hao Huang , Yi Zhang","doi":"10.1016/j.jcta.2026.106163","DOIUrl":"10.1016/j.jcta.2026.106163","url":null,"abstract":"<div><div>A family of subsets <span><math><mi>F</mi></math></span> is intersecting if <span><math><mi>A</mi><mo>∩</mo><mi>B</mi><mo>≠</mo><mo>∅</mo></math></span> for any <span><math><mi>A</mi><mo>,</mo><mi>B</mi><mo>∈</mo><mi>F</mi></math></span>. In this paper, we show that for given integers <span><math><mi>k</mi><mo>></mo><mi>d</mi><mo>≥</mo><mn>2</mn></math></span> and <span><math><mi>n</mi><mo>≥</mo><mn>2</mn><mi>k</mi><mo>+</mo><mn>2</mn><mi>d</mi><mo>−</mo><mn>3</mn></math></span>, and any intersecting family <span><math><mi>F</mi></math></span> of <em>k</em>-subsets of <span><math><mo>{</mo><mn>1</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>n</mi><mo>}</mo></math></span>, there exists a <em>d</em>-subset of <span><math><mo>[</mo><mi>n</mi><mo>]</mo></math></span> contained in at most <span><math><mo>(</mo><mtable><mtr><mtd><mrow><mi>n</mi><mo>−</mo><mi>d</mi><mo>−</mo><mn>1</mn></mrow></mtd></mtr><mtr><mtd><mrow><mi>k</mi><mo>−</mo><mi>d</mi><mo>−</mo><mn>1</mn></mrow></mtd></mtr></mtable><mo>)</mo></math></span> subsets of <span><math><mi>F</mi></math></span>. This result, proved using spectral graph theory, gives a <em>d</em>-degree generalization of the celebrated Erdős–Ko–Rado Theorem, improving a theorem of Kupavskii.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"221 ","pages":"Article 106163"},"PeriodicalIF":1.2,"publicationDate":"2026-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146048526","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-07-01Epub Date: 2025-12-22DOI: 10.1016/j.jcta.2025.106156
Chengyang Qian, Yaokun Wu, Yinfeng Zhu
For a positive integer t, a t-variable digraph on a set K is defined as a map f from to . Note that an ordinary digraph is a 1-variable digraph. In 2017, Wu, Xu, and Zhu proposed the study of multivariate digraphs as a qualitative counterpart of going from Markov chains to higher-order Markov chains. A fundamental parameter of any strongly connected ordinary digraph is its period. This notion extends naturally to strongly connected t-variable digraphs. Let denote the set of all possible periods of strongly connected t-variable digraphs, let be its Frobenius number (i.e., the largest nonnegative integer not belonging to ), and let be its Sylvester number (i.e., the number of positive integers outside of ). In this paper, we establish new estimates for and . We also show that equals when and {1} when . Although this work originated in an effort to understand qualitative higher-order Markov chains, it turns out to be closely related to two other active areas of research, partitioning a discrete box into subboxes, and restricted universal partial cycles.
{"title":"Periods of strongly connected multivariate digraphs","authors":"Chengyang Qian, Yaokun Wu, Yinfeng Zhu","doi":"10.1016/j.jcta.2025.106156","DOIUrl":"10.1016/j.jcta.2025.106156","url":null,"abstract":"<div><div>For a positive integer <em>t</em>, a <em>t</em>-variable digraph on a set <em>K</em> is defined as a map <em>f</em> from <span><math><msup><mrow><mi>K</mi></mrow><mrow><mi>t</mi></mrow></msup></math></span> to <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>K</mi></mrow></msup></math></span>. Note that an ordinary digraph is a 1-variable digraph. In 2017, Wu, Xu, and Zhu proposed the study of multivariate digraphs as a qualitative counterpart of going from Markov chains to higher-order Markov chains. A fundamental parameter of any strongly connected ordinary digraph is its period. This notion extends naturally to strongly connected <em>t</em>-variable digraphs. Let <span><math><mrow><mi>PS</mi></mrow><mo>(</mo><mi>t</mi><mo>)</mo></math></span> denote the set of all possible periods of strongly connected <em>t</em>-variable digraphs, let <span><math><mi>g</mi><mo>(</mo><mi>t</mi><mo>)</mo></math></span> be its Frobenius number (i.e., the largest nonnegative integer not belonging to <span><math><mrow><mi>PS</mi></mrow><mo>(</mo><mi>t</mi><mo>)</mo></math></span>), and let <span><math><mi>n</mi><mo>(</mo><mi>t</mi><mo>)</mo></math></span> be its Sylvester number (i.e., the number of positive integers outside of <span><math><mrow><mi>PS</mi></mrow><mo>(</mo><mi>t</mi><mo>)</mo></math></span>). In this paper, we establish new estimates for <span><math><mi>g</mi><mo>(</mo><mi>t</mi><mo>)</mo></math></span> and <span><math><mi>n</mi><mo>(</mo><mi>t</mi><mo>)</mo></math></span>. We also show that <span><math><mrow><mi>PS</mi></mrow><mo>(</mo><mi>t</mi><mo>)</mo><mo>∩</mo><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mn>4</mn><mi>t</mi><mo>−</mo><mn>1</mn><mo>}</mo></math></span> equals <span><math><mo>{</mo><mn>1</mn><mo>,</mo><mn>8</mn><mo>}</mo></math></span> when <span><math><mi>t</mi><mo>∈</mo><mo>{</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>}</mo></math></span> and {1} when <span><math><mi>t</mi><mo>≥</mo><mn>5</mn></math></span>. Although this work originated in an effort to understand qualitative higher-order Markov chains, it turns out to be closely related to two other active areas of research, partitioning a discrete box into subboxes, and restricted universal partial cycles.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"221 ","pages":"Article 106156"},"PeriodicalIF":1.2,"publicationDate":"2026-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145813851","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-07-01Epub Date: 2026-01-22DOI: 10.1016/j.jcta.2026.106159
Qipin Chen , Shane Chern , Atsuro Yoshida
Koutschan, Krattenthaler and Schlosser recently considered a family of binomial determinants. In this work, we give combinatorial interpretations of two subclasses of these determinants in terms of domino tilings and nonintersecting lattice paths, thereby partially answering a question of theirs. Furthermore, the determinant evaluations established by Koutschan, Krattenthaler and Schlosser produce many product formulas for our weighted enumerations of domino tilings and nonintersecting lattice paths. However, there are still two enumerations left corresponding to conjectural formulas made by the three. We hereby prove the two conjectures using the principle of holonomic Ansatz plus the approach of modular reduction for creative telescoping, and hence fill the gap.
{"title":"Domino tilings, nonintersecting lattice paths and subclasses of Koutschan–Krattenthaler–Schlosser determinants","authors":"Qipin Chen , Shane Chern , Atsuro Yoshida","doi":"10.1016/j.jcta.2026.106159","DOIUrl":"10.1016/j.jcta.2026.106159","url":null,"abstract":"<div><div>Koutschan, Krattenthaler and Schlosser recently considered a family of binomial determinants. In this work, we give combinatorial interpretations of two subclasses of these determinants in terms of domino tilings and nonintersecting lattice paths, thereby partially answering a question of theirs. Furthermore, the determinant evaluations established by Koutschan, Krattenthaler and Schlosser produce many product formulas for our weighted enumerations of domino tilings and nonintersecting lattice paths. However, there are still two enumerations left corresponding to conjectural formulas made by the three. We hereby prove the two conjectures using the principle of holonomic Ansatz plus the approach of modular reduction for creative telescoping, and hence fill the gap.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"221 ","pages":"Article 106159"},"PeriodicalIF":1.2,"publicationDate":"2026-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146023005","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Solutions to some problems on unique representation bases","authors":"Yuchen Ding","doi":"10.1016/j.jcta.2026.106166","DOIUrl":"10.1016/j.jcta.2026.106166","url":null,"abstract":"<div><div>In this note, three 2003 problems of Nathanson and two 2007 problems of Chen on unique representation bases for the integers are resolved.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"221 ","pages":"Article 106166"},"PeriodicalIF":1.2,"publicationDate":"2026-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146072029","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-07-01Epub Date: 2026-01-23DOI: 10.1016/j.jcta.2026.106165
Kaimei Huang , Zhicong Lin , Sherry H.F. Yan
A pair of permutation statistics is said to be r-Euler-Mahonian over multipermutations if and , are equidistributed over the set of all multipermutations of M for any given multiset M, where denotes the r-descent number and denotes the r-major index introduced by Rawlings. In this paper, we shall introduce the r-gap excedance number and the r-gap Denert's statistic for multipermutations and prove that is r-Euler-Mahonian over multipermutations, thereby extending Liu's result on permutations to multipermutations. When , our result recovers the equidistribution of and over derived by Han.
{"title":"On r-Euler-Mahonian statistics for multipermutations","authors":"Kaimei Huang , Zhicong Lin , Sherry H.F. Yan","doi":"10.1016/j.jcta.2026.106165","DOIUrl":"10.1016/j.jcta.2026.106165","url":null,"abstract":"<div><div>A pair <span><math><mo>(</mo><msub><mrow><mi>st</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>st</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span> of permutation statistics is said to be <em>r</em>-Euler-Mahonian over multipermutations if <span><math><mo>(</mo><msub><mrow><mi>st</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>st</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span> and <span><math><mo>(</mo><mi>r</mi><mrow><mi>des</mi></mrow></math></span>, <span><math><mi>r</mi><mrow><mi>maj</mi></mrow><mo>)</mo></math></span> are equidistributed over the set <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>M</mi></mrow></msub></math></span> of all multipermutations of <em>M</em> for any given multiset <em>M</em>, where <span><math><mi>r</mi><mrow><mi>des</mi></mrow></math></span> denotes the <em>r</em>-descent number and <span><math><mi>r</mi><mrow><mi>maj</mi></mrow></math></span> denotes the <em>r</em>-major index introduced by Rawlings. In this paper, we shall introduce the <em>r</em>-gap excedance number <span><math><mi>r</mi><mrow><mi>exc</mi></mrow></math></span> and the <em>r</em>-gap Denert's statistic <span><math><mi>r</mi><mrow><mi>den</mi></mrow></math></span> for multipermutations and prove that <span><math><mo>(</mo><mi>r</mi><mrow><mi>exc</mi></mrow><mo>,</mo><mi>r</mi><mrow><mi>den</mi></mrow><mo>)</mo></math></span> is <em>r</em>-Euler-Mahonian over multipermutations, thereby extending Liu's result on permutations to multipermutations. When <span><math><mi>r</mi><mo>=</mo><mn>1</mn></math></span>, our result recovers the equidistribution of <span><math><mo>(</mo><mrow><mi>des</mi></mrow><mo>,</mo><mrow><mi>maj</mi></mrow><mo>)</mo></math></span> and <span><math><mo>(</mo><mrow><mi>exc</mi></mrow><mo>,</mo><mrow><mi>den</mi></mrow><mo>)</mo></math></span> over <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>M</mi></mrow></msub></math></span> derived by Han.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"221 ","pages":"Article 106165"},"PeriodicalIF":1.2,"publicationDate":"2026-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146023006","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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