Pub Date : 2026-01-01Epub Date: 2025-06-03DOI: 10.1016/j.jcta.2025.106076
Quan-Hui Yang , Lilu Zhao
It is proved that for all sufficiently large positive integers n, if with and , then there exists a power of 2 which can be represented as the sum of at most 22 distinct elements of A. This answers a question in [13]. The result is optimal in two aspects. In the above conclusion, the condition cannot be replaced by , and the number 22 is also best possible.
{"title":"Optimal result on restricted sumsets containing powers of two","authors":"Quan-Hui Yang , Lilu Zhao","doi":"10.1016/j.jcta.2025.106076","DOIUrl":"10.1016/j.jcta.2025.106076","url":null,"abstract":"<div><div>It is proved that for all sufficiently large positive integers <em>n</em>, if <span><math><mi>A</mi><mo>⊆</mo><mo>[</mo><mn>1</mn><mo>,</mo><mi>n</mi><mo>]</mo></math></span> with <span><math><mo>|</mo><mi>A</mi><mo>|</mo><mo>></mo><mfrac><mrow><mi>n</mi></mrow><mrow><mn>6</mn></mrow></mfrac><mo>+</mo><mn>2</mn></math></span> and <span><math><mrow><mi>gcd</mi><mspace></mspace></mrow><mi>A</mi><mo>=</mo><mn>1</mn></math></span>, then there exists a power of 2 which can be represented as the sum of at most 22 distinct elements of <em>A</em>. This answers a question in <span><span>[13]</span></span>. The result is optimal in two aspects. In the above conclusion, the condition <span><math><mo>|</mo><mi>A</mi><mo>|</mo><mo>></mo><mfrac><mrow><mi>n</mi></mrow><mrow><mn>6</mn></mrow></mfrac><mo>+</mo><mn>2</mn></math></span> cannot be replaced by <span><math><mo>|</mo><mi>A</mi><mo>|</mo><mo>></mo><mfrac><mrow><mi>n</mi></mrow><mrow><mn>6</mn></mrow></mfrac><mo>+</mo><mn>1</mn></math></span>, and the number 22 is also best possible.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"217 ","pages":"Article 106076"},"PeriodicalIF":0.9,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144204299","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 : 2026-01-01Epub Date: 2025-07-22DOI: 10.1016/j.jcta.2025.106094
Sun Kim
Ramanujan's modular equations are closely associated with partition identities. In particular, the modular equations of prime degrees , 23 and the corresponding partition identities are of very elegant forms. These five modular equations were extensively generalized by Warnaar and the present author in the form of general theta function identities. In this paper, we provide further general theta function identities and present many partition identities as special cases.
{"title":"General Theta function identities","authors":"Sun Kim","doi":"10.1016/j.jcta.2025.106094","DOIUrl":"10.1016/j.jcta.2025.106094","url":null,"abstract":"<div><div>Ramanujan's modular equations are closely associated with partition identities. In particular, the modular equations of prime degrees <span><math><mn>3</mn><mo>,</mo><mn>5</mn><mo>,</mo><mn>7</mn><mo>,</mo><mn>11</mn></math></span>, 23 and the corresponding partition identities are of very elegant forms. These five modular equations were extensively generalized by Warnaar and the present author in the form of general theta function identities. In this paper, we provide further general theta function identities and present many partition identities as special cases.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"217 ","pages":"Article 106094"},"PeriodicalIF":0.9,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144679155","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 : 2026-01-01Epub Date: 2025-07-11DOI: 10.1016/j.jcta.2025.106093
Shi-Mei Ma , Hao Qi , Jean Yeh , Yeong-Nan Yeh
In the context of Stirling polynomials, Gessel and Stanley introduced Stirling permutations, which have attracted extensive attention over the past decades. Recently, we introduced Stirling permutation codes and provided numerous equidistribution results as applications. The purpose of the present work is to further analyze Stirling permutation codes. First, we derive an expansion formula expressing the joint distribution of the types A and B descent statistics over the hyperoctahedral group, and we also find an interlacing property involving the zeros of its coefficient polynomials. Next, we prove a strong connection between signed permutations in the hyperoctahedral group and Stirling permutations. We also study unified generalizations of the trivariate second-order Eulerian and ascent-plateau polynomials. Using Stirling permutation codes, we provide expansion formulas for eight-variable and seventeen-variable polynomials, which imply several e-positive expansions and clarify the connection among several statistics. Our results generalize the results of Bóna, Chen-Fu, Dumont, Haglund-Visontai, Janson and Petersen.
{"title":"Stirling permutation codes. II","authors":"Shi-Mei Ma , Hao Qi , Jean Yeh , Yeong-Nan Yeh","doi":"10.1016/j.jcta.2025.106093","DOIUrl":"10.1016/j.jcta.2025.106093","url":null,"abstract":"<div><div>In the context of Stirling polynomials, Gessel and Stanley introduced Stirling permutations, which have attracted extensive attention over the past decades. Recently, we introduced Stirling permutation codes and provided numerous equidistribution results as applications. The purpose of the present work is to further analyze Stirling permutation codes. First, we derive an expansion formula expressing the joint distribution of the types <em>A</em> and <em>B</em> descent statistics over the hyperoctahedral group, and we also find an interlacing property involving the zeros of its coefficient polynomials. Next, we prove a strong connection between signed permutations in the hyperoctahedral group and Stirling permutations. We also study unified generalizations of the trivariate second-order Eulerian and ascent-plateau polynomials. Using Stirling permutation codes, we provide expansion formulas for eight-variable and seventeen-variable polynomials, which imply several <em>e</em>-positive expansions and clarify the connection among several statistics. Our results generalize the results of Bóna, Chen-Fu, Dumont, Haglund-Visontai, Janson and Petersen.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"217 ","pages":"Article 106093"},"PeriodicalIF":0.9,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144596379","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 : 2026-01-01Epub Date: 2025-06-27DOI: 10.1016/j.jcta.2025.106090
Thomas Britz , Himadri Shekhar Chakraborty , Tsuyoshi Miezaki
In this paper, we present the harmonic generalizations of well-known polynomials of codes over finite fields, namely the higher weight enumerators and the extended weight enumerators, and we derive the correspondences between these weight enumerators. Moreover, we present the harmonic generalization of Greene's Theorem for the higher (resp. extended) weight enumerators. As an application of this Greene's-type theorem, we provide the MacWilliams-type identity for harmonic higher weight enumerators of codes over finite fields. Finally, we use this new identity to give a new proof of the Assmus-Mattson Theorem for subcode supports of linear codes over finite fields using harmonic higher weight enumerators.
{"title":"Harmonic higher and extended weight enumerators","authors":"Thomas Britz , Himadri Shekhar Chakraborty , Tsuyoshi Miezaki","doi":"10.1016/j.jcta.2025.106090","DOIUrl":"10.1016/j.jcta.2025.106090","url":null,"abstract":"<div><div>In this paper, we present the harmonic generalizations of well-known polynomials of codes over finite fields, namely the higher weight enumerators and the extended weight enumerators, and we derive the correspondences between these weight enumerators. Moreover, we present the harmonic generalization of Greene's Theorem for the higher (resp. extended) weight enumerators. As an application of this Greene's-type theorem, we provide the MacWilliams-type identity for harmonic higher weight enumerators of codes over finite fields. Finally, we use this new identity to give a new proof of the Assmus-Mattson Theorem for subcode supports of linear codes over finite fields using harmonic higher weight enumerators.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"217 ","pages":"Article 106090"},"PeriodicalIF":0.9,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144489510","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 : 2025-11-01Epub Date: 2025-05-29DOI: 10.1016/j.jcta.2025.106075
Xin Wei , Xiande Zhang , Gennian Ge
Separating hash families are useful combinatorial structures which generalize several well-studied objects in cryptography and coding theory. Let denote the maximum size of universe for a t-perfect hash family of length N over an alphabet of size q. In this paper, we show that for all , which answers an open problem about separating hash families raised by Blackburn et al. in 2008 for certain parameters. Previously, this result was known only for . Our proof is obtained by establishing the existence of a large set of integers avoiding nontrivial solutions to a set of correlated linear equations.
{"title":"Separating hash families with large universe","authors":"Xin Wei , Xiande Zhang , Gennian Ge","doi":"10.1016/j.jcta.2025.106075","DOIUrl":"10.1016/j.jcta.2025.106075","url":null,"abstract":"<div><div>Separating hash families are useful combinatorial structures which generalize several well-studied objects in cryptography and coding theory. Let <span><math><msub><mrow><mi>p</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>(</mo><mi>N</mi><mo>,</mo><mi>q</mi><mo>)</mo></math></span> denote the maximum size of universe for a <em>t</em>-perfect hash family of length <em>N</em> over an alphabet of size <em>q</em>. In this paper, we show that <span><math><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn><mo>−</mo><mi>o</mi><mo>(</mo><mn>1</mn><mo>)</mo></mrow></msup><mo><</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><mi>q</mi><mo>)</mo><mo>=</mo><mi>o</mi><mo>(</mo><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span> for all <span><math><mi>t</mi><mo>≥</mo><mn>3</mn></math></span>, which answers an open problem about separating hash families raised by Blackburn et al. in 2008 for certain parameters. Previously, this result was known only for <span><math><mi>t</mi><mo>=</mo><mn>3</mn><mo>,</mo><mn>4</mn></math></span>. Our proof is obtained by establishing the existence of a large set of integers avoiding nontrivial solutions to a set of correlated linear equations.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"216 ","pages":"Article 106075"},"PeriodicalIF":0.9,"publicationDate":"2025-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144167274","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 : 2025-11-01Epub 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":"2025-11-01","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 : 2025-11-01Epub 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":"2025-11-01","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 : 2025-11-01Epub 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":"2025-11-01","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 : 2025-11-01Epub 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":"2025-11-01","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 : 2025-11-01Epub Date: 2025-05-22DOI: 10.1016/j.jcta.2025.106065
Antonio Montero , Primož Potočnik
The motion of a graph is the minimum number of vertices that are moved by a non-trivial automorphism. Equivalently, it can be defined as the minimal degree of its automorphism group (as a permutation group on the vertices). In this paper, we develop some results on permutation groups (primitive and imprimitive) with small minimal degree. As a consequence of such results, we classify vertex-transitive graphs whose motion is 4 or a prime number.
{"title":"Vertex-transitive graphs with small motion and transitive permutation groups with small minimal degree","authors":"Antonio Montero , Primož Potočnik","doi":"10.1016/j.jcta.2025.106065","DOIUrl":"10.1016/j.jcta.2025.106065","url":null,"abstract":"<div><div>The motion of a graph is the minimum number of vertices that are moved by a non-trivial automorphism. Equivalently, it can be defined as the minimal degree of its automorphism group (as a permutation group on the vertices). In this paper, we develop some results on permutation groups (primitive and imprimitive) with small minimal degree. As a consequence of such results, we classify vertex-transitive graphs whose motion is 4 or a prime number.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"216 ","pages":"Article 106065"},"PeriodicalIF":0.9,"publicationDate":"2025-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144106349","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}
扫码关注我们
求助内容：
应助结果提醒方式：
京公网安备 11010802042870号