Pub Date : 2025-05-02DOI: 10.1016/j.jcta.2025.106063
Jianrong Zhao , Chenxu Wang , Yu Fu
<div><div>Let <span><math><mi>S</mi><mo>=</mo><mo>{</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>}</mo></math></span> be a gcd-closed set (i.e. <span><math><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>)</mo><mo>∈</mo><mi>S</mi></math></span> for all <span><math><mn>1</mn><mo>≤</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>≤</mo><mi>n</mi></math></span>). In 2002, Hong proposed the divisibility problem of characterizing all gcd-closed sets <em>S</em> with <span><math><mo>|</mo><mi>S</mi><mo>|</mo><mo>≥</mo><mn>4</mn></math></span> such that the GCD matrix (<em>S</em>) divides the LCM matrix <span><math><mo>[</mo><mi>S</mi><mo>]</mo></math></span> in the ring <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>Z</mi><mo>)</mo></math></span>. For <span><math><mi>x</mi><mo>∈</mo><mi>S</mi></math></span>, let <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>S</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>:</mo><mo>=</mo><mo>{</mo><mi>z</mi><mo>∈</mo><mi>S</mi><mo>:</mo><mi>z</mi><mo><</mo><mi>x</mi><mo>,</mo><mi>z</mi><mo>|</mo><mi>x</mi><mtext> and </mtext><mo>(</mo><mi>z</mi><mo>|</mo><mi>y</mi><mo>|</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><mi>S</mi><mo>)</mo><mo>⇒</mo><mi>y</mi><mo>∈</mo><mo>{</mo><mi>z</mi><mo>,</mo><mi>x</mi><mo>}</mo><mo>}</mo></math></span>. In 2009, Feng, Hong and Zhao answered this problem in the context where <span><math><msub><mrow><mi>max</mi></mrow><mrow><mi>x</mi><mo>∈</mo><mi>S</mi></mrow></msub><mo></mo><mo>{</mo><mo>|</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>S</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>|</mo><mo>}</mo><mo>≤</mo><mn>2</mn></math></span>. In 2022, Zhao, Chen and Hong obtained a necessary and sufficient condition on the gcd-closed set <em>S</em> with <span><math><msub><mrow><mi>max</mi></mrow><mrow><mi>x</mi><mo>∈</mo><mi>S</mi></mrow></msub><mo></mo><mo>{</mo><mo>|</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>S</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>|</mo><mo>}</mo><mo>=</mo><mn>3</mn></math></span> such that <span><math><mo>(</mo><mi>S</mi><mo>)</mo><mo>|</mo><mrow><mo>[</mo><mi>S</mi><mo>]</mo></mrow></math></span>. Meanwhile, they raised a conjecture on the necessary and sufficient condition such that <span><math><mo>(</mo><mi>S</mi><mo>)</mo><mo>|</mo><mrow><mo>[</mo><mi>S</mi><mo>]</mo></mrow></math></span> holds for the remaining case <span><math><msub><mrow><mi>max</mi></mrow><mrow><mi>x</mi><mo>∈</mo><mi>S</mi></mrow></msub><mo></mo><mo>{</mo><mo>|</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>S</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>|</mo><mo>}</mo><mo>≥</mo><mn>4</mn></math></span>. In this paper, we confirm the Zhao-Chen-Hong conjecture from a novel perspective, consequently solve Hong's open problem completely.</d
设S={x1，…，xn}是一个gcd闭集（即(xi,xj)∈S，对于所有1≤i，j≤n）。2002年，Hong提出了表征所有GCD -闭集S的|S|≥4使得GCD矩阵(S)能除环Mn(Z)中的LCM矩阵[S]的可分性问题。x∈年代,让GS (x): = {z∈年代:z< x, z | x和y z | | x, y∈(S)⇒y∈{z、x}}。2009年，Feng， Hong和Zhao在maxx∈S∈{|GS(x)|}≤2的情况下回答了这个问题。Zhao、Chen和Hong在2022年得到了maxx∈S∈S (|GS(x)|}=3的gcd-闭集S上的一个充要条件，使得(S)|[S]。同时，他们提出了一个关于(S)|[S]对剩余情况maxx∈S∈{|GS(x)|}≥4成立的充分必要条件的猜想。本文从一个全新的角度证实了赵-陈-洪猜想，从而彻底解决了洪的开放问题。
{"title":"Studying the divisibility of power LCM matrices by power GCD matrices on gcd-closed sets","authors":"Jianrong Zhao , Chenxu Wang , Yu Fu","doi":"10.1016/j.jcta.2025.106063","DOIUrl":"10.1016/j.jcta.2025.106063","url":null,"abstract":"<div><div>Let <span><math><mi>S</mi><mo>=</mo><mo>{</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>}</mo></math></span> be a gcd-closed set (i.e. <span><math><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>)</mo><mo>∈</mo><mi>S</mi></math></span> for all <span><math><mn>1</mn><mo>≤</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>≤</mo><mi>n</mi></math></span>). In 2002, Hong proposed the divisibility problem of characterizing all gcd-closed sets <em>S</em> with <span><math><mo>|</mo><mi>S</mi><mo>|</mo><mo>≥</mo><mn>4</mn></math></span> such that the GCD matrix (<em>S</em>) divides the LCM matrix <span><math><mo>[</mo><mi>S</mi><mo>]</mo></math></span> in the ring <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>Z</mi><mo>)</mo></math></span>. For <span><math><mi>x</mi><mo>∈</mo><mi>S</mi></math></span>, let <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>S</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>:</mo><mo>=</mo><mo>{</mo><mi>z</mi><mo>∈</mo><mi>S</mi><mo>:</mo><mi>z</mi><mo><</mo><mi>x</mi><mo>,</mo><mi>z</mi><mo>|</mo><mi>x</mi><mtext> and </mtext><mo>(</mo><mi>z</mi><mo>|</mo><mi>y</mi><mo>|</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><mi>S</mi><mo>)</mo><mo>⇒</mo><mi>y</mi><mo>∈</mo><mo>{</mo><mi>z</mi><mo>,</mo><mi>x</mi><mo>}</mo><mo>}</mo></math></span>. In 2009, Feng, Hong and Zhao answered this problem in the context where <span><math><msub><mrow><mi>max</mi></mrow><mrow><mi>x</mi><mo>∈</mo><mi>S</mi></mrow></msub><mo></mo><mo>{</mo><mo>|</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>S</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>|</mo><mo>}</mo><mo>≤</mo><mn>2</mn></math></span>. In 2022, Zhao, Chen and Hong obtained a necessary and sufficient condition on the gcd-closed set <em>S</em> with <span><math><msub><mrow><mi>max</mi></mrow><mrow><mi>x</mi><mo>∈</mo><mi>S</mi></mrow></msub><mo></mo><mo>{</mo><mo>|</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>S</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>|</mo><mo>}</mo><mo>=</mo><mn>3</mn></math></span> such that <span><math><mo>(</mo><mi>S</mi><mo>)</mo><mo>|</mo><mrow><mo>[</mo><mi>S</mi><mo>]</mo></mrow></math></span>. Meanwhile, they raised a conjecture on the necessary and sufficient condition such that <span><math><mo>(</mo><mi>S</mi><mo>)</mo><mo>|</mo><mrow><mo>[</mo><mi>S</mi><mo>]</mo></mrow></math></span> holds for the remaining case <span><math><msub><mrow><mi>max</mi></mrow><mrow><mi>x</mi><mo>∈</mo><mi>S</mi></mrow></msub><mo></mo><mo>{</mo><mo>|</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>S</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>|</mo><mo>}</mo><mo>≥</mo><mn>4</mn></math></span>. In this paper, we confirm the Zhao-Chen-Hong conjecture from a novel perspective, consequently solve Hong's open problem completely.</d","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"215 ","pages":"Article 106063"},"PeriodicalIF":0.9,"publicationDate":"2025-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143895609","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 : 2025-04-30DOI: 10.1016/j.jcta.2025.106061
Bence Csajbók , Zoltán Lóránt Nagy
A subset S of an abelian group G is called 3-AP free if it does not contain a three term arithmetic progression. Moreover, S is called complete 3-AP free, if it is maximal w.r.t. set inclusion. One of the most central problems in additive combinatorics is to determine the maximal size of a 3-AP free set, which is necessarily complete. In this paper we are interested in the minimum size of complete 3-AP free sets. We define and study saturation w.r.t. 3-APs and present constructions of small complete 3-AP free sets and 3-AP saturating sets for several families of vector spaces and cyclic groups.
{"title":"Complete 3-term arithmetic progression free sets of small size in vector spaces and other abelian groups","authors":"Bence Csajbók , Zoltán Lóránt Nagy","doi":"10.1016/j.jcta.2025.106061","DOIUrl":"10.1016/j.jcta.2025.106061","url":null,"abstract":"<div><div>A subset <em>S</em> of an abelian group <em>G</em> is called 3-AP free if it does not contain a three term arithmetic progression. Moreover, <em>S</em> is called complete 3-AP free, if it is maximal w.r.t. set inclusion. One of the most central problems in additive combinatorics is to determine the maximal size of a 3-AP free set, which is necessarily complete. In this paper we are interested in the minimum size of complete 3-AP free sets. We define and study saturation w.r.t. 3-APs and present constructions of small complete 3-AP free sets and 3-AP saturating sets for several families of vector spaces and cyclic groups.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"215 ","pages":"Article 106061"},"PeriodicalIF":0.9,"publicationDate":"2025-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143886651","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 : 2025-04-30DOI: 10.1016/j.jcta.2025.106060
Aleksei L. Perezhogin , Vladimir N. Potapov , Anna A. Taranenko , Sergey Yu. Vladimirov
A multidimensional nonnegative matrix is called polystochastic if the sum of its entries over each line is equal to 1. The permanent of a multidimensional matrix is the sum of products of entries over all diagonals. We prove that if d is even, then the permanent of a d-dimensional polystochastic matrix of order 4 is positive, and for odd d, we give a complete characterization of d-dimensional polystochastic matrices with zero permanent.
{"title":"Characterization of polystochastic matrices of order 4 with zero permanent","authors":"Aleksei L. Perezhogin , Vladimir N. Potapov , Anna A. Taranenko , Sergey Yu. Vladimirov","doi":"10.1016/j.jcta.2025.106060","DOIUrl":"10.1016/j.jcta.2025.106060","url":null,"abstract":"<div><div>A multidimensional nonnegative matrix is called polystochastic if the sum of its entries over each line is equal to 1. The permanent of a multidimensional matrix is the sum of products of entries over all diagonals. We prove that if <em>d</em> is even, then the permanent of a <em>d</em>-dimensional polystochastic matrix of order 4 is positive, and for odd <em>d</em>, we give a complete characterization of <em>d</em>-dimensional polystochastic matrices with zero permanent.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"215 ","pages":"Article 106060"},"PeriodicalIF":0.9,"publicationDate":"2025-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143886650","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 : 2025-04-29DOI: 10.1016/j.jcta.2025.106064
Sho Kubota , Kiyoto Yoshino
We completely characterize circulant graphs with valency up to 4 that admit perfect state transfer. Those of valency 3 do not admit it. On the other hand, circulant graphs with valency 4 admit perfect state transfer only in two infinite families: one discovered by Zhan and another new family, while no others do. The main tools for deriving these results are symmetry of graphs and eigenvalues. We describe necessary conditions for perfect state transfer to occur based on symmetry of graphs, which mathematically refers to automorphisms of graphs. As for eigenvalues, if perfect state transfer occurs, then certain eigenvalues of the corresponding isotropic random walks must be the halves of algebraic integers. Taking this into account, we utilize known results on the rings of integers of cyclotomic fields.
{"title":"Circulant graphs with valency up to 4 that admit perfect state transfer in Grover walks","authors":"Sho Kubota , Kiyoto Yoshino","doi":"10.1016/j.jcta.2025.106064","DOIUrl":"10.1016/j.jcta.2025.106064","url":null,"abstract":"<div><div>We completely characterize circulant graphs with valency up to 4 that admit perfect state transfer. Those of valency 3 do not admit it. On the other hand, circulant graphs with valency 4 admit perfect state transfer only in two infinite families: one discovered by Zhan and another new family, while no others do. The main tools for deriving these results are symmetry of graphs and eigenvalues. We describe necessary conditions for perfect state transfer to occur based on symmetry of graphs, which mathematically refers to automorphisms of graphs. As for eigenvalues, if perfect state transfer occurs, then certain eigenvalues of the corresponding isotropic random walks must be the halves of algebraic integers. Taking this into account, we utilize known results on the rings of integers of cyclotomic fields.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"216 ","pages":"Article 106064"},"PeriodicalIF":0.9,"publicationDate":"2025-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143882707","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 : 2025-04-29DOI: 10.1016/j.jcta.2025.106062
Yongjiang Wu, Lihua Feng, Yongtao Li
<div><div>Two families <span><math><mi>F</mi></math></span> and <span><math><mi>G</mi></math></span> are called cross-intersecting if for every <span><math><mi>F</mi><mo>∈</mo><mi>F</mi></math></span> and <span><math><mi>G</mi><mo>∈</mo><mi>G</mi></math></span>, the intersection <span><math><mi>F</mi><mo>∩</mo><mi>G</mi></math></span> is non-empty. For any positive integers <em>n</em> and <em>k</em>, let <span><math><mo>(</mo><mtable><mtr><mtd><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow></mtd></mtr><mtr><mtd><mi>k</mi></mtd></mtr></mtable><mo>)</mo></math></span> denote the family of all <em>k</em>-element subsets of <span><math><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo>}</mo></math></span>. Let <span><math><mi>t</mi><mo>,</mo><mi>s</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>n</mi></math></span> be non-negative integers with <span><math><mi>k</mi><mo>≥</mo><mi>s</mi><mo>+</mo><mn>1</mn></math></span> and <span><math><mi>n</mi><mo>≥</mo><mn>2</mn><mi>k</mi><mo>+</mo><mi>t</mi></math></span>. In 2016, Frankl proved that if <span><math><mi>F</mi><mo>⊆</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow></mtd></mtr><mtr><mtd><mrow><mi>k</mi><mo>+</mo><mi>t</mi></mrow></mtd></mtr></mtable><mo>)</mo></mrow></math></span> and <span><math><mi>G</mi><mo>⊆</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow></mtd></mtr><mtr><mtd><mi>k</mi></mtd></mtr></mtable><mo>)</mo></mrow></math></span> are cross-intersecting families, and <span><math><mi>F</mi></math></span> is <span><math><mo>(</mo><mi>t</mi><mo>+</mo><mn>1</mn><mo>)</mo></math></span>-intersecting and <span><math><mo>|</mo><mi>F</mi><mo>|</mo><mo>≥</mo><mn>1</mn></math></span>, then <span><math><mo>|</mo><mi>F</mi><mo>|</mo><mo>+</mo><mo>|</mo><mi>G</mi><mo>|</mo><mo>≤</mo><mrow><mo>(</mo><mtable><mtr><mtd><mi>n</mi></mtd></mtr><mtr><mtd><mi>k</mi></mtd></mtr></mtable><mo>)</mo></mrow><mo>−</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mi>n</mi><mo>−</mo><mi>k</mi><mo>−</mo><mi>t</mi></mrow></mtd></mtr><mtr><mtd><mi>k</mi></mtd></mtr></mtable><mo>)</mo></mrow><mo>+</mo><mn>1</mn></math></span>. Furthermore, Frankl conjectured that under an additional condition <span><math><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mo>[</mo><mi>k</mi><mo>+</mo><mi>t</mi><mo>+</mo><mi>s</mi><mo>]</mo></mrow></mtd></mtr><mtr><mtd><mrow><mi>k</mi><mo>+</mo><mi>t</mi></mrow></mtd></mtr></mtable><mo>)</mo></mrow><mo>⊆</mo><mi>F</mi></math></span>, the following inequality holds:<span><span><span><math><mo>|</mo><mi>F</mi><mo>|</mo><mo>+</mo><mo>|</mo><mi>G</mi><mo>|</mo><mo>≤</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mi>k</mi><mo>+</mo><mi>t</mi><mo>+</mo><mi>s</mi></mrow></mtd></mtr><mtr><mtd><mrow><mi>k</mi><mo>+</mo><mi>t</mi></mrow></mtd></mtr></mtable><mo>)</mo></mrow><mo>+</mo><mrow><mo>(</mo><mtable><mtr><mtd><mi>n</mi></mtd></mtr><mtr><mtd><mi>k</mi></mtd></mtr></mtable><mo>)</mo></mrow><mo>−</mo><munderover><mo>∑</mo><mr
{"title":"Proof of Frankl's conjecture on cross-intersecting families","authors":"Yongjiang Wu, Lihua Feng, Yongtao Li","doi":"10.1016/j.jcta.2025.106062","DOIUrl":"10.1016/j.jcta.2025.106062","url":null,"abstract":"<div><div>Two families <span><math><mi>F</mi></math></span> and <span><math><mi>G</mi></math></span> are called cross-intersecting if for every <span><math><mi>F</mi><mo>∈</mo><mi>F</mi></math></span> and <span><math><mi>G</mi><mo>∈</mo><mi>G</mi></math></span>, the intersection <span><math><mi>F</mi><mo>∩</mo><mi>G</mi></math></span> is non-empty. For any positive integers <em>n</em> and <em>k</em>, let <span><math><mo>(</mo><mtable><mtr><mtd><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow></mtd></mtr><mtr><mtd><mi>k</mi></mtd></mtr></mtable><mo>)</mo></math></span> denote the family of all <em>k</em>-element subsets of <span><math><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo>}</mo></math></span>. Let <span><math><mi>t</mi><mo>,</mo><mi>s</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>n</mi></math></span> be non-negative integers with <span><math><mi>k</mi><mo>≥</mo><mi>s</mi><mo>+</mo><mn>1</mn></math></span> and <span><math><mi>n</mi><mo>≥</mo><mn>2</mn><mi>k</mi><mo>+</mo><mi>t</mi></math></span>. In 2016, Frankl proved that if <span><math><mi>F</mi><mo>⊆</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow></mtd></mtr><mtr><mtd><mrow><mi>k</mi><mo>+</mo><mi>t</mi></mrow></mtd></mtr></mtable><mo>)</mo></mrow></math></span> and <span><math><mi>G</mi><mo>⊆</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow></mtd></mtr><mtr><mtd><mi>k</mi></mtd></mtr></mtable><mo>)</mo></mrow></math></span> are cross-intersecting families, and <span><math><mi>F</mi></math></span> is <span><math><mo>(</mo><mi>t</mi><mo>+</mo><mn>1</mn><mo>)</mo></math></span>-intersecting and <span><math><mo>|</mo><mi>F</mi><mo>|</mo><mo>≥</mo><mn>1</mn></math></span>, then <span><math><mo>|</mo><mi>F</mi><mo>|</mo><mo>+</mo><mo>|</mo><mi>G</mi><mo>|</mo><mo>≤</mo><mrow><mo>(</mo><mtable><mtr><mtd><mi>n</mi></mtd></mtr><mtr><mtd><mi>k</mi></mtd></mtr></mtable><mo>)</mo></mrow><mo>−</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mi>n</mi><mo>−</mo><mi>k</mi><mo>−</mo><mi>t</mi></mrow></mtd></mtr><mtr><mtd><mi>k</mi></mtd></mtr></mtable><mo>)</mo></mrow><mo>+</mo><mn>1</mn></math></span>. Furthermore, Frankl conjectured that under an additional condition <span><math><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mo>[</mo><mi>k</mi><mo>+</mo><mi>t</mi><mo>+</mo><mi>s</mi><mo>]</mo></mrow></mtd></mtr><mtr><mtd><mrow><mi>k</mi><mo>+</mo><mi>t</mi></mrow></mtd></mtr></mtable><mo>)</mo></mrow><mo>⊆</mo><mi>F</mi></math></span>, the following inequality holds:<span><span><span><math><mo>|</mo><mi>F</mi><mo>|</mo><mo>+</mo><mo>|</mo><mi>G</mi><mo>|</mo><mo>≤</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mi>k</mi><mo>+</mo><mi>t</mi><mo>+</mo><mi>s</mi></mrow></mtd></mtr><mtr><mtd><mrow><mi>k</mi><mo>+</mo><mi>t</mi></mrow></mtd></mtr></mtable><mo>)</mo></mrow><mo>+</mo><mrow><mo>(</mo><mtable><mtr><mtd><mi>n</mi></mtd></mtr><mtr><mtd><mi>k</mi></mtd></mtr></mtable><mo>)</mo></mrow><mo>−</mo><munderover><mo>∑</mo><mr","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"216 ","pages":"Article 106062"},"PeriodicalIF":0.9,"publicationDate":"2025-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143882709","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 : 2025-04-25DOI: 10.1016/j.jcta.2025.106059
Ishay Haviv
The storage capacity of a graph measures the maximum amount of information that can be stored across its vertices, such that the information at any vertex can be recovered from the information stored at its neighborhood. The study of this graph quantity is motivated by applications in distributed storage and by its intimate relations to the index coding problem from the area of network information theory. In the latter, one wishes to minimize the amount of information that has to be transmitted to a collection of receivers, in a way that enables each of them to discover its required data using some prior side information.
In this paper, we initiate the study of the and problems from the perspective of parameterized complexity. We prove that the problem parameterized by the solution size admits a kernelization algorithm producing kernels of linear size. We also provide such a result for the problem, in the linear and non-linear settings, where it is parameterized by the dual value of the solution, i.e., the length of the transmission that can be saved using the side information. A key ingredient in the proofs is the crown decomposition technique due to Chor, Fellows, and Juedes [14], [11]. As an application, we significantly extend an algorithmic result of Dau, Skachek, and Chee [13].
{"title":"Kernels for storage capacity and dual index coding","authors":"Ishay Haviv","doi":"10.1016/j.jcta.2025.106059","DOIUrl":"10.1016/j.jcta.2025.106059","url":null,"abstract":"<div><div>The storage capacity of a graph measures the maximum amount of information that can be stored across its vertices, such that the information at any vertex can be recovered from the information stored at its neighborhood. The study of this graph quantity is motivated by applications in distributed storage and by its intimate relations to the index coding problem from the area of network information theory. In the latter, one wishes to minimize the amount of information that has to be transmitted to a collection of receivers, in a way that enables each of them to discover its required data using some prior side information.</div><div>In this paper, we initiate the study of the <figure><img></figure> and <figure><img></figure> problems from the perspective of parameterized complexity. We prove that the <figure><img></figure> problem parameterized by the solution size admits a kernelization algorithm producing kernels of linear size. We also provide such a result for the <figure><img></figure> problem, in the linear and non-linear settings, where it is parameterized by the dual value of the solution, i.e., the length of the transmission that can be saved using the side information. A key ingredient in the proofs is the crown decomposition technique due to Chor, Fellows, and Juedes <span><span>[14]</span></span>, <span><span>[11]</span></span>. As an application, we significantly extend an algorithmic result of Dau, Skachek, and Chee <span><span>[13]</span></span>.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"216 ","pages":"Article 106059"},"PeriodicalIF":0.9,"publicationDate":"2025-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143868647","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 : 2025-04-16DOI: 10.1016/j.jcta.2025.106051
Yuefeng Yang , Akihiro Munemasa , Kaishun Wang , Wenying Zhu
We classify certain non-symmetric commutative association schemes. As an application, we determine all the weakly distance-regular circulants of one type of arcs by using Schur rings. We also give the classification of primitive weakly distance-regular circulants.
{"title":"Weakly distance-regular circulants, I","authors":"Yuefeng Yang , Akihiro Munemasa , Kaishun Wang , Wenying Zhu","doi":"10.1016/j.jcta.2025.106051","DOIUrl":"10.1016/j.jcta.2025.106051","url":null,"abstract":"<div><div>We classify certain non-symmetric commutative association schemes. As an application, we determine all the weakly distance-regular circulants of one type of arcs by using Schur rings. We also give the classification of primitive weakly distance-regular circulants.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"216 ","pages":"Article 106051"},"PeriodicalIF":0.9,"publicationDate":"2025-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143834212","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 : 2025-04-03DOI: 10.1016/j.jcta.2025.106047
Dean Crnković , Maarten De Boeck , Francesco Pavese , Andrea Švob
Divisible design graphs were introduced in 2011 by Haemers, Kharaghani and Meulenberg. In this paper, we introduce the notion of q-analogs of divisible design graphs and show that all q-analogs of divisible design graphs come from spreads, and are actually q-analogs of strongly regular graphs.
Deza graphs were introduced by Erickson, Fernando, Haemers, Hardy and Hemmeter in 1999. In this paper, we introduce q-analogs of Deza graphs. Further, we determine possible parameters, give examples of q-analogs of Deza graphs and characterize all non-strongly regular q-analogs of Deza graphs with the smallest parameters.
{"title":"q-Analogs of divisible design graphs and Deza graphs","authors":"Dean Crnković , Maarten De Boeck , Francesco Pavese , Andrea Švob","doi":"10.1016/j.jcta.2025.106047","DOIUrl":"10.1016/j.jcta.2025.106047","url":null,"abstract":"<div><div>Divisible design graphs were introduced in 2011 by Haemers, Kharaghani and Meulenberg. In this paper, we introduce the notion of <em>q</em>-analogs of divisible design graphs and show that all <em>q</em>-analogs of divisible design graphs come from spreads, and are actually <em>q</em>-analogs of strongly regular graphs.</div><div>Deza graphs were introduced by Erickson, Fernando, Haemers, Hardy and Hemmeter in 1999. In this paper, we introduce <em>q</em>-analogs of Deza graphs. Further, we determine possible parameters, give examples of <em>q</em>-analogs of Deza graphs and characterize all non-strongly regular <em>q</em>-analogs of Deza graphs with the smallest parameters.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"215 ","pages":"Article 106047"},"PeriodicalIF":0.9,"publicationDate":"2025-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143761107","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-04-03DOI: 10.1016/j.jcta.2025.106049
Zhicong Lin , Jing Liu , Sherry H.F. Yan
Motivated by Kitaev and Zhang's recent work on non-overlapping ascents in stack-sortable permutations and Dumont's permutation interpretation of the Jacobi elliptic functions, we investigate some parity statistics on restricted permutations. Some new related bijections are constructed and two refinements of the generating function for descents over 321-avoiding permutations due to Barnabei, Bonetti and Silimbanian are obtained. In particular, an open problem of Kitaev and Zhang about non-overlapping ascents on 321-avoiding permutations is solved and several combinatorial interpretations for the Catalan–Schett polynomials are found. The stack-sortable permutations are at the heart of our approaches.
{"title":"Parity statistics on restricted permutations and the Catalan–Schett polynomials","authors":"Zhicong Lin , Jing Liu , Sherry H.F. Yan","doi":"10.1016/j.jcta.2025.106049","DOIUrl":"10.1016/j.jcta.2025.106049","url":null,"abstract":"<div><div>Motivated by Kitaev and Zhang's recent work on non-overlapping ascents in stack-sortable permutations and Dumont's permutation interpretation of the Jacobi elliptic functions, we investigate some parity statistics on restricted permutations. Some new related bijections are constructed and two refinements of the generating function for descents over 321-avoiding permutations due to Barnabei, Bonetti and Silimbanian are obtained. In particular, an open problem of Kitaev and Zhang about non-overlapping ascents on 321-avoiding permutations is solved and several combinatorial interpretations for the Catalan–Schett polynomials are found. The stack-sortable permutations are at the heart of our approaches.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"215 ","pages":"Article 106049"},"PeriodicalIF":0.9,"publicationDate":"2025-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143761108","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 : 2025-04-03DOI: 10.1016/j.jcta.2025.106046
Zhiguo Ding , Michael E. Zieve
For each odd prime power q, we describe a class of rational functions with the following unusual property: for every odd j, the function induced by on is 2-to-1. We also show that, among all known rational functions which are 2-to-1 on for infinitely many j, our new functions are the only ones which cannot be written as compositions of rational functions of degree at most four, monomials, Dickson polynomials, and additive (linearized) polynomials.
{"title":"Exceptional 2-to-1 rational functions","authors":"Zhiguo Ding , Michael E. Zieve","doi":"10.1016/j.jcta.2025.106046","DOIUrl":"10.1016/j.jcta.2025.106046","url":null,"abstract":"<div><div>For each odd prime power <em>q</em>, we describe a class of rational functions <span><math><mi>f</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math></span> with the following unusual property: for every odd <em>j</em>, the function induced by <span><math><mi>f</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span> on <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>j</mi></mrow></msup></mrow></msub><mo>∪</mo><mo>{</mo><mo>∞</mo><mo>}</mo></math></span> is 2-to-1. We also show that, among all known rational functions <span><math><mi>f</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math></span> which are 2-to-1 on <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>j</mi></mrow></msup></mrow></msub><mo>∪</mo><mo>{</mo><mo>∞</mo><mo>}</mo></math></span> for infinitely many <em>j</em>, our new functions are the only ones which cannot be written as compositions of rational functions of degree at most four, monomials, Dickson polynomials, and additive (linearized) polynomials.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"215 ","pages":"Article 106046"},"PeriodicalIF":0.9,"publicationDate":"2025-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143761106","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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