Combinatorica最新文献

{"title":"Counting Perfect Matchings in Dirac Hypergraphs","authors":"Matthew Kwan, Roodabeh Safavi, Yiting Wang","doi":"10.1007/s00493-025-00194-8","DOIUrl":"https://doi.org/10.1007/s00493-025-00194-8","url":null,"abstract":"One of the foundational theorems of extremal graph theory is <jats:italic>Dirac’s theorem</jats:italic> , which says that if an <jats:italic>n</jats:italic> -vertex graph <jats:italic>G</jats:italic> has minimum degree at least <jats:italic>n</jats:italic> /2, then <jats:italic>G</jats:italic> has a Hamilton cycle, and therefore a perfect matching (if <jats:italic>n</jats:italic> is even). Later work by Sárközy, Selkow and Szemerédi showed that in fact Dirac graphs have <jats:italic>many</jats:italic> Hamilton cycles and perfect matchings, culminating in a result of Cuckler and Kahn that gives a precise description of the numbers of Hamilton cycles and perfect matchings in a Dirac graph <jats:italic>G</jats:italic> (in terms of an entropy-like parameter of <jats:italic>G</jats:italic> ). In this paper we extend Cuckler and Kahn’s result to perfect matchings in hypergraphs. For positive integers <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$d<k$$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo><</mml:mo> <mml:mi>k</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> , and for <jats:italic>n</jats:italic> divisible by <jats:italic>k</jats:italic> , let <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$m_{d}(k,n)$$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:msub> <mml:mi>m</mml:mi> <mml:mi>d</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>k</mml:mi> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> be the minimum <jats:italic>d</jats:italic> -degree that ensures the existence of a perfect matching in an <jats:italic>n</jats:italic> -vertex <jats:italic>k</jats:italic> -uniform hypergraph. In general, it is an open question to determine (even asymptotically) the values of <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$m_{d}(k,n)$$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:msub> <mml:mi>m</mml:mi> <mml:mi>d</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>k</mml:mi> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> , but we are nonetheless able to prove an analogue of the Cuckler–Kahn theorem, showing that if an <jats:italic>n</jats:italic> -vertex <jats:italic>k</jats:italic> -uniform hypergraph <jats:italic>G</jats:italic> has minimum <jats:italic>d</jats:italic> -degree at least <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$(1+gamma )m_{d}(k,n)$$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:mi>γ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:msub> <mml:mi>m</mml:mi> <mml:mi>d</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"117 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2026-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146070700","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":"Additive Structure in Convex Sets","authors":"Thomas F. Bloom, Jakob Führer, Oliver Roche-Newton","doi":"10.1007/s00493-026-00200-7","DOIUrl":"https://doi.org/10.1007/s00493-026-00200-7","url":null,"abstract":"This paper considers some different measures for how additively structured a convex set can be. The main result gives a construction of a convex set <jats:italic>A</jats:italic> containing <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$Omega (|A|^{3/2})$$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>Ω</mml:mi> <mml:mo>(</mml:mo> <mml:mo>|</mml:mo> <mml:mi>A</mml:mi> <mml:msup> <mml:mo>|</mml:mo> <mml:mrow> <mml:mn>3</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> three-term arithmetic progressions.","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"388 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2026-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146070554","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":"Topology of Independence Complexes and Cycle Structure of Hypergraphs","authors":"Jinha Kim","doi":"10.1007/s00493-026-00198-y","DOIUrl":"https://doi.org/10.1007/s00493-026-00198-y","url":null,"abstract":"","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"238 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2026-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146070555","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":"Quotient-Convergence of Submodular Setfunctions","authors":"Kristóf Bérczi, Márton Borbényi, László Lovász, László Márton Tóth","doi":"10.1007/s00493-026-00199-x","DOIUrl":"https://doi.org/10.1007/s00493-026-00199-x","url":null,"abstract":"We introduce the concept of quotient-convergence for sequences of submodular set functions, providing, among others, a new framework for the study of convergence of matroids through their rank functions. Extending the limit theory of bounded degree graphs, which analyzes graph sequences via neighborhood sampling, we address the challenge posed by the absence of a neighborhood concept in matroids. We show that any bounded set function can be approximated by a sequence of finite set functions that quotient-converges to it. In addition, we explicitly construct such sequences for increasing, submodular, and upper continuous set functions, and prove the completeness of the space under quotient-convergence.","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"179 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2026-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146070553","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":"Semiregular Abstract Polyhedra with Trivial Facet Stabilizer","authors":"Elías Mochán","doi":"10.1007/s00493-025-00197-5","DOIUrl":"https://doi.org/10.1007/s00493-025-00197-5","url":null,"abstract":"<jats:italic> polytopes</jats:italic> generalize the face lattice of convex polytopes. A polytope is <jats:italic>semiregular</jats:italic> if its facets are regular and its automorphism group acts transitively on its vertices. In this paper we construct semiregular, facet-transitive polyhedra with trivial facet stabilizer, showing that semiregular abstract polyhedra can have an unbounded number of flag orbits, while having as little as one facet orbit. We interpret this construction in terms of operations applied to high rank regular and chiral polytopes, and we see how these same operations help us construct alternating semiregular polyhedra (that is, with two facet orbits and adjacent facets in different orbits). Finally, we give an idea to generalize this construction giving examples in higher ranks.","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"85 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2026-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145947443","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":"Good Bounds for Sets Lacking Skew Corners","authors":"Luka Milićević","doi":"10.1007/s00493-025-00196-6","DOIUrl":"https://doi.org/10.1007/s00493-025-00196-6","url":null,"abstract":"","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"29 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2026-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145947442","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":"Multiplicative irreducibility of small perturbations of the set of shifted k-th powers","authors":"Chi Hoi Yip","doi":"10.1007/s00493-025-00193-9","DOIUrl":"https://doi.org/10.1007/s00493-025-00193-9","url":null,"abstract":"Motivated by a conjecture of Erdős on the additive irreducibility of small perturbations of the set of squares, recently Hajdu and Sárközy studied a multiplicative analogue of the conjecture for shifted <jats:italic>k</jats:italic> -th powers. They conjectured that for each <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$kge 2$$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> , if one changes <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$o(X^{1/k})$$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>o</mml:mi> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>X</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mi>k</mml:mi> </mml:mrow> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> elements of <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$M_k'={x^k+1: x in mathbb {N}}$$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:msubsup> <mml:mi>M</mml:mi> <mml:mi>k</mml:mi> <mml:mo>′</mml:mo> </mml:msubsup> <mml:mo>=</mml:mo> <mml:mrow> <mml:mo>{</mml:mo> <mml:msup> <mml:mi>x</mml:mi> <mml:mi>k</mml:mi> </mml:msup> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo>:</mml:mo> <mml:mi>x</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>N</mml:mi> <mml:mo>}</mml:mo> </mml:mrow> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> up to <jats:italic>X</jats:italic> , then the resulting set cannot be written as a product set <jats:italic>AB</jats:italic> nontrivially. In this paper, we confirm a more general version of their conjecture for <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$kge 3$$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> .","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"6 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2026-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145947395","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":"The complex Illumination problem","authors":"Liran Rotem, Alon Schejter, Boaz A. Slomka","doi":"10.1007/s00493-025-00195-7","DOIUrl":"https://doi.org/10.1007/s00493-025-00195-7","url":null,"abstract":"We formulate a complex analog of the celebrated Levi-Hadwiger-Boltyanski illumination (or covering) conjecture for complex convex bodies in <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$mathbb {C}^n$$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msup> <mml:mrow> <mml:mi>C</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:math> </jats:alternatives> </jats:inline-formula> , as well as its (non-comparable) fractional version. A key element in posing these problems is computing the classical and fractional illumination numbers of the complex analog of the hypercube, i.e., the polydisc. We prove that the illumination number of the polydisc in <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$mathbb {C}^n$$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msup> <mml:mrow> <mml:mi>C</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:math> </jats:alternatives> </jats:inline-formula> is equal to <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$2^{n+1}-1$$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:msup> <mml:mn>2</mml:mn> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> and that the fractional illumination number of the polydisc in <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$mathbb {C}^n$$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msup> <mml:mrow> <mml:mi>C</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:math> </jats:alternatives> </jats:inline-formula> is equal to <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$2^n$$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msup> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> </mml:msup> </mml:math> </jats:alternatives> </jats:inline-formula> . In addition, we verify both conjectures for the classes of complex zonotopes and zonoids.","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"17 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2026-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145947444","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":"p-Anisotropy on the Moment Curve for Homology Manifolds and Cycles","authors":"Karim Alexander Adiprasito, Kaiying Hou, Daishi Kiyohara, Daniel Koizumi, Monroe Stephenson","doi":"10.1007/s00493-025-00192-w","DOIUrl":"https://doi.org/10.1007/s00493-025-00192-w","url":null,"abstract":"","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"166 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2025-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145753139","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":"Bound on Shortest Cycle Covers","authors":"Deping Song, Xuding Zhu","doi":"10.1007/s00493-025-00191-x","DOIUrl":"https://doi.org/10.1007/s00493-025-00191-x","url":null,"abstract":"","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"28 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2025-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145753129","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}