Pub Date : 2025-07-11DOI: 10.1016/j.ejc.2025.104217
Martin Balko , Anna Brötzner , Fabian Klute , Josef Tkadlec
We initiate the study of extremal problems about faces in convex rectilinear drawings of , that is, drawings where vertices are represented by points in the plane in convex position and edges by line segments between the points representing the end-vertices. We show that if a convex rectilinear drawing of does not contain a common interior point of at least three edges, then there is always a face forming a convex 5-gon while there are such drawings without any face forming a convex -gon with .
A convex rectilinear drawing of is regular if its vertices correspond to vertices of a regular convex -gon. We characterize positive integers for which regular drawings of contain a face forming a convex 5-gon.
To our knowledge, this type of problems has not been considered in the literature before and so we also pose several new natural open problems.
{"title":"Faces in rectilinear drawings of complete graphs","authors":"Martin Balko , Anna Brötzner , Fabian Klute , Josef Tkadlec","doi":"10.1016/j.ejc.2025.104217","DOIUrl":"10.1016/j.ejc.2025.104217","url":null,"abstract":"<div><div>We initiate the study of extremal problems about faces in <em>convex rectilinear drawings</em> of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, that is, drawings where vertices are represented by points in the plane in convex position and edges by line segments between the points representing the end-vertices. We show that if a convex rectilinear drawing of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> does not contain a common interior point of at least three edges, then there is always a face forming a convex 5-gon while there are such drawings without any face forming a convex <span><math><mi>k</mi></math></span>-gon with <span><math><mrow><mi>k</mi><mo>≥</mo><mn>6</mn></mrow></math></span>.</div><div>A convex rectilinear drawing of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> is <em>regular</em> if its vertices correspond to vertices of a regular convex <span><math><mi>n</mi></math></span>-gon. We characterize positive integers <span><math><mi>n</mi></math></span> for which regular drawings of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> contain a face forming a convex 5-gon.</div><div>To our knowledge, this type of problems has not been considered in the literature before and so we also pose several new natural open problems.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"130 ","pages":"Article 104217"},"PeriodicalIF":1.0,"publicationDate":"2025-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144596135","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-07-11DOI: 10.1016/j.ejc.2025.104219
Kağan Kurşungöz, Halı̇me Ömrüuzun Seyrek
We introduce the notion of pivot in a chain of skew diagrams in the context of cylindric partitions. Then, we show that cylindric partitions are in one-to-one correspondence with a pair consisting of an ordinary partition and a suitably restricted chain of pivots. Next, we show the general form of the generating function for cylindric partitions into distinct parts and give some examples. We prove part of a conjecture by Corteel, Dousse, and Uncu. The approaches and proofs are elementary and combinatorial.
{"title":"A decomposition of cylindric partitions and cylindric partitions into distinct parts","authors":"Kağan Kurşungöz, Halı̇me Ömrüuzun Seyrek","doi":"10.1016/j.ejc.2025.104219","DOIUrl":"10.1016/j.ejc.2025.104219","url":null,"abstract":"<div><div>We introduce the notion of <em>pivot</em> in a chain of skew diagrams in the context of cylindric partitions. Then, we show that cylindric partitions are in one-to-one correspondence with a pair consisting of an ordinary partition and a suitably restricted chain of pivots. Next, we show the general form of the generating function for cylindric partitions into distinct parts and give some examples. We prove part of a conjecture by Corteel, Dousse, and Uncu. The approaches and proofs are elementary and combinatorial.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"130 ","pages":"Article 104219"},"PeriodicalIF":1.0,"publicationDate":"2025-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144605306","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-07-09DOI: 10.1016/j.ejc.2025.104222
Lintong Wang, Sherry H.F. Yan
A partially ordered pattern (abbreviated POP) is a partially ordered set (poset) that generalizes the notion of a pattern when we are not concerned with the relative order of some of its letters. The notion of partially ordered patterns provides a convenient language to deal with large sets of permutation patterns. In analogy to the shape-Wilf-equivalence for permutation patterns, Burstein–Han–Kitaev–Zhang initiated the study of the shape-Wilf-equivalence for POPs which would result in the shape-Wilf-equivalence for large sets of permutation patterns. The main objective of this paper is to confirm a recent intriguing conjecture posed by Burstein–Han–Kitaev–Zhang concerning the shape-Wilf-equivalence for POPs of length . This is accomplished by establishing a bijection between two sets of pattern-avoiding transversals of a given Young diagram.
{"title":"Proof of a conjecture on the shape-Wilf-equivalence for partially ordered patterns","authors":"Lintong Wang, Sherry H.F. Yan","doi":"10.1016/j.ejc.2025.104222","DOIUrl":"10.1016/j.ejc.2025.104222","url":null,"abstract":"<div><div>A partially ordered pattern (abbreviated POP) is a partially ordered set (poset) that generalizes the notion of a pattern when we are not concerned with the relative order of some of its letters. The notion of partially ordered patterns provides a convenient language to deal with large sets of permutation patterns. In analogy to the shape-Wilf-equivalence for permutation patterns, Burstein–Han–Kitaev–Zhang initiated the study of the shape-Wilf-equivalence for POPs which would result in the shape-Wilf-equivalence for large sets of permutation patterns. The main objective of this paper is to confirm a recent intriguing conjecture posed by Burstein–Han–Kitaev–Zhang concerning the shape-Wilf-equivalence for POPs of length <span><math><mi>k</mi></math></span>. This is accomplished by establishing a bijection between two sets of pattern-avoiding transversals of a given Young diagram.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"130 ","pages":"Article 104222"},"PeriodicalIF":1.0,"publicationDate":"2025-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144579676","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-07-09DOI: 10.1016/j.ejc.2025.104221
Zhiyun Cheng
Recently, Chmutov introduced the partial duality of ribbon graphs, which can be regarded as a generalization of the classical Euler-Poincaré duality. The partial-dual genus polynomial is an enumeration of the partial duals of by Euler genus. For an intersection graph derived from a given chord diagram, the partial-dual genus polynomial can be defined by considering the ribbon graph associated to the chord diagram. In this paper, we provide a combinatorial approach to the partial-dual genus polynomial in terms of intersection graphs without referring to chord diagrams. After extending the definition of the partial-dual genus polynomial from intersection graphs to all graphs, we prove that it satisfies the four-term relation of graphs. This provides an answer to a problem proposed by Chmutov (2023).
{"title":"Partial-dual genus polynomial of graphs","authors":"Zhiyun Cheng","doi":"10.1016/j.ejc.2025.104221","DOIUrl":"10.1016/j.ejc.2025.104221","url":null,"abstract":"<div><div>Recently, Chmutov introduced the partial duality of ribbon graphs, which can be regarded as a generalization of the classical Euler-Poincaré duality. The partial-dual genus polynomial <span><math><mrow><msup><mrow></mrow><mrow><mi>∂</mi></mrow></msup><msub><mrow><mi>ɛ</mi></mrow><mrow><mi>G</mi></mrow></msub><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></math></span> is an enumeration of the partial duals of <span><math><mi>G</mi></math></span> by Euler genus. For an intersection graph derived from a given chord diagram, the partial-dual genus polynomial can be defined by considering the ribbon graph associated to the chord diagram. In this paper, we provide a combinatorial approach to the partial-dual genus polynomial in terms of intersection graphs without referring to chord diagrams. After extending the definition of the partial-dual genus polynomial from intersection graphs to all graphs, we prove that it satisfies the four-term relation of graphs. This provides an answer to a problem proposed by Chmutov (2023).</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"130 ","pages":"Article 104221"},"PeriodicalIF":1.0,"publicationDate":"2025-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144579677","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-07-08DOI: 10.1016/j.ejc.2025.104213
Seung-Il Choi , Sun-Young Nam , Young-Tak Oh
In this paper, we explore the relationship between quasisymmetric Schur -functions and peak Young quasisymmetric Schur functions. We introduce a bijection on such that and share identical descent distributions. Here, is the set of standard peak immaculate tableaux of shape , and and denote column reading and row reading, respectively. By combining this equidistribution with the algorithm developed by Allen, Hallam, and Mason, we demonstrate that the transition matrix from the basis of quasisymmetric Schur -functions to the basis of peak Young quasisymmetric Schur functions is upper triangular, with entries being non-negative integers. Furthermore, we provide explicit descriptions of the expansion of peak Young quasisymmetric Schur functions in specific cases, in terms of quasisymmetric Schur -functions. We also investigate the combinatorial properties of standard peak immaculate tableaux, standard Young composition tableaux, and standard peak Young composition tableaux. We provide a hook length formula for and show that standard Young composition tableaux and standard peak Young composition tableaux can be each bijectively mapped to words satisfying suitable conditions. Especially, cases of compositions with rectangular shape are examined in detail.
{"title":"Quasisymmetric Schur Q-functions and peak Young quasisymmetric Schur functions","authors":"Seung-Il Choi , Sun-Young Nam , Young-Tak Oh","doi":"10.1016/j.ejc.2025.104213","DOIUrl":"10.1016/j.ejc.2025.104213","url":null,"abstract":"<div><div>In this paper, we explore the relationship between quasisymmetric Schur <span><math><mi>Q</mi></math></span>-functions and peak Young quasisymmetric Schur functions. We introduce a bijection on <span><math><mrow><mi>SPIT</mi><mrow><mo>(</mo><mi>α</mi><mo>)</mo></mrow></mrow></math></span> such that <span><math><mrow><mo>{</mo><msub><mrow><mi>w</mi></mrow><mrow><mi>c</mi></mrow></msub><mrow><mo>(</mo><mi>T</mi><mo>)</mo></mrow><mo>∣</mo><mi>T</mi><mo>∈</mo><mi>SPIT</mi><mrow><mo>(</mo><mi>α</mi><mo>)</mo></mrow><mo>}</mo></mrow></math></span> and <span><math><mrow><mo>{</mo><msub><mrow><mi>w</mi></mrow><mrow><mi>r</mi></mrow></msub><mrow><mo>(</mo><mi>T</mi><mo>)</mo></mrow><mo>∣</mo><mi>T</mi><mo>∈</mo><mi>SPIT</mi><mrow><mo>(</mo><mi>α</mi><mo>)</mo></mrow><mo>}</mo></mrow></math></span> share identical descent distributions. Here, <span><math><mrow><mi>SPIT</mi><mrow><mo>(</mo><mi>α</mi><mo>)</mo></mrow></mrow></math></span> is the set of standard peak immaculate tableaux of shape <span><math><mi>α</mi></math></span>, and <span><math><msub><mrow><mi>w</mi></mrow><mrow><mi>c</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>w</mi></mrow><mrow><mi>r</mi></mrow></msub></math></span> denote column reading and row reading, respectively. By combining this equidistribution with the algorithm developed by Allen, Hallam, and Mason, we demonstrate that the transition matrix from the basis of quasisymmetric Schur <span><math><mi>Q</mi></math></span>-functions to the basis of peak Young quasisymmetric Schur functions is upper triangular, with entries being non-negative integers. Furthermore, we provide explicit descriptions of the expansion of peak Young quasisymmetric Schur functions in specific cases, in terms of quasisymmetric Schur <span><math><mi>Q</mi></math></span>-functions. We also investigate the combinatorial properties of standard peak immaculate tableaux, standard Young composition tableaux, and standard peak Young composition tableaux. We provide a hook length formula for <span><math><mrow><mi>SPIT</mi><mrow><mo>(</mo><mi>α</mi><mo>)</mo></mrow></mrow></math></span> and show that standard Young composition tableaux and standard peak Young composition tableaux can be each bijectively mapped to words satisfying suitable conditions. Especially, cases of compositions with rectangular shape are examined in detail.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"130 ","pages":"Article 104213"},"PeriodicalIF":1.0,"publicationDate":"2025-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144571712","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-07-08DOI: 10.1016/j.ejc.2025.104214
Sebastian Mies , Benjamin Moore , Evelyne Smith-Roberge
The pseudoforest version of the Strong Nine Dragon Tree Conjecture states that if a graph has maximum average degree at most , then it has a decomposition into pseudoforests where in one pseudoforest the components of have at most edges. This was proven in 2020 in Grout and Moore (2020). We strengthen this theorem by showing that we can find such a decomposition where additionally is acyclic, the diameter of the components of is at most , where , and at most if . Furthermore, for any component of and any , we have if . We also show that both diameter bounds are best possible as an extension for both the Strong Nine Dragon Tree Conjecture for pseudoforests and its original conjecture for forests. In fact, they are still optimal even if we only enforce to have any constant maximum degree, instead of enforcing every component of to have at most edges.
{"title":"Beyond the pseudoforest strong Nine Dragon Tree Theorem","authors":"Sebastian Mies , Benjamin Moore , Evelyne Smith-Roberge","doi":"10.1016/j.ejc.2025.104214","DOIUrl":"10.1016/j.ejc.2025.104214","url":null,"abstract":"<div><div>The pseudoforest version of the Strong Nine Dragon Tree Conjecture states that if a graph <span><math><mi>G</mi></math></span> has maximum average degree <span><math><mrow><mi>m</mi><mi>a</mi><mi>d</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mn>2</mn><msub><mrow><mo>max</mo></mrow><mrow><mi>H</mi><mo>⊆</mo><mi>G</mi></mrow></msub><mfrac><mrow><mi>e</mi><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow></mrow><mrow><mi>v</mi><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow></mrow></mfrac></mrow></math></span> at most <span><math><mrow><mn>2</mn><mrow><mo>(</mo><mi>k</mi><mo>+</mo><mfrac><mrow><mi>d</mi></mrow><mrow><mi>d</mi><mo>+</mo><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mo>)</mo></mrow></mrow></math></span>, then it has a decomposition into <span><math><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></math></span> pseudoforests where in one pseudoforest <span><math><mi>F</mi></math></span> the components of <span><math><mi>F</mi></math></span> have at most <span><math><mi>d</mi></math></span> edges. This was proven in 2020 in Grout and Moore (2020). We strengthen this theorem by showing that we can find such a decomposition where additionally <span><math><mi>F</mi></math></span> is acyclic, the diameter of the components of <span><math><mi>F</mi></math></span> is at most <span><math><mrow><mn>2</mn><mi>ℓ</mi><mo>+</mo><mn>2</mn></mrow></math></span>, where <span><math><mrow><mi>ℓ</mi><mo>=</mo><mfenced><mrow><mfrac><mrow><mi>d</mi><mo>−</mo><mn>1</mn></mrow><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mrow></mfenced></mrow></math></span>, and at most <span><math><mrow><mn>2</mn><mi>ℓ</mi><mo>+</mo><mn>1</mn></mrow></math></span> if <span><math><mrow><mi>d</mi><mo>≡</mo><mn>1</mn><mspace></mspace><mo>mod</mo><mspace></mspace><mrow><mo>(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow></mrow></math></span>. Furthermore, for any component <span><math><mi>K</mi></math></span> of <span><math><mi>F</mi></math></span> and any <span><math><mrow><mi>z</mi><mo>∈</mo><mi>N</mi></mrow></math></span>, we have <span><math><mrow><mi>d</mi><mi>i</mi><mi>a</mi><mi>m</mi><mrow><mo>(</mo><mi>K</mi><mo>)</mo></mrow><mo>≤</mo><mn>2</mn><mi>z</mi></mrow></math></span> if <span><math><mrow><mi>e</mi><mrow><mo>(</mo><mi>K</mi><mo>)</mo></mrow><mo>≥</mo><mi>d</mi><mo>−</mo><mi>z</mi><mrow><mo>(</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mo>+</mo><mn>1</mn></mrow></math></span>. We also show that both diameter bounds are best possible as an extension for both the Strong Nine Dragon Tree Conjecture for pseudoforests and its original conjecture for forests. In fact, they are still optimal even if we only enforce <span><math><mi>F</mi></math></span> to have any constant maximum degree, instead of enforcing every component of <span><math><mi>F</mi></math></span> to have at most <span><math><mi>d</mi></math></span> edges.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"130 ","pages":"Article 104214"},"PeriodicalIF":1.0,"publicationDate":"2025-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144579675","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-07-04DOI: 10.1016/j.ejc.2025.104212
Panna Gehér , Géza Tóth
A matchstick graph is a plane graph with edges drawn as unit distance line segments. This class of graphs was introduced by Harborth who conjectured that a matchstick graph on vertices can have at most edges. Recently, his conjecture was settled by Lavollée and Swanepoel. In this paper we consider 1-planar unit distance graphs. We say that a graph is a 1-planar unit distance graph if it can be drawn in the plane such that all edges are drawn as unit distance line segments while each of them are involved in at most one crossing. We show that such graphs on vertices can have at most edges, which is almost tight. We also investigate some generalizations, namely -planar and -quasiplanar unit distance graphs.
{"title":"1-planar unit distance graphs","authors":"Panna Gehér , Géza Tóth","doi":"10.1016/j.ejc.2025.104212","DOIUrl":"10.1016/j.ejc.2025.104212","url":null,"abstract":"<div><div>A matchstick graph is a plane graph with edges drawn as unit distance line segments. This class of graphs was introduced by Harborth who conjectured that a matchstick graph on <span><math><mi>n</mi></math></span> vertices can have at most <span><math><mrow><mo>⌊</mo><mn>3</mn><mi>n</mi><mo>−</mo><msqrt><mrow><mn>12</mn><mi>n</mi><mo>−</mo><mn>3</mn></mrow></msqrt><mo>⌋</mo></mrow></math></span> edges. Recently, his conjecture was settled by Lavollée and Swanepoel. In this paper we consider 1-planar unit distance graphs. We say that a graph is a 1-planar unit distance graph if it can be drawn in the plane such that all edges are drawn as unit distance line segments while each of them are involved in at most one crossing. We show that such graphs on <span><math><mi>n</mi></math></span> vertices can have at most <span><math><mrow><mn>3</mn><mi>n</mi><mo>−</mo><mroot><mrow><mi>n</mi></mrow><mrow><mn>4</mn></mrow></mroot><mo>/</mo><mn>15</mn></mrow></math></span> edges, which is almost tight. We also investigate some generalizations, namely <span><math><mi>k</mi></math></span>-planar and <span><math><mi>k</mi></math></span>-quasiplanar unit distance graphs.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"130 ","pages":"Article 104212"},"PeriodicalIF":1.0,"publicationDate":"2025-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144549398","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-07-01DOI: 10.1016/j.ejc.2025.104201
Michael A. Henning , Anders Yeo
<div><div>The transversal number <span><math><mrow><mi>τ</mi><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow></mrow></math></span> of a hypergraph <span><math><mi>H</mi></math></span> is the minimum number of vertices that intersect every edge of <span><math><mi>H</mi></math></span>. A 6-uniform hypergraph has all edges of size 6. On 10 November 2000 Tuza and Vestergaard (2002) conjectured that if <span><math><mi>H</mi></math></span> is a 3-regular 6-uniform hypergraph of order <span><math><mi>n</mi></math></span>, then <span><math><mrow><mi>τ</mi><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow><mo>≤</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac><mi>n</mi></mrow></math></span>. This conjecture was recently proven by the Henning and Yeo (2023) and is now called the Tuza-Vestergaard Theorem. In this paper we extend the Tuza-Vestergaard Theorem by relaxing the 3-regularity constraint and allowing bounded maximum degree 4. We present several applications of the Tuza-Vestergaard Theorem and its extension. We obtain best known upper bounds to date on the transversal number of a (general) 6-uniform hypergraph <span><math><mi>H</mi></math></span> of order <span><math><mi>n</mi></math></span> and size <span><math><mi>m</mi></math></span>. In particular, if <span><math><mi>H</mi></math></span> is a 4-regular 6-uniform hypergraph of order <span><math><mi>n</mi></math></span>, then we show that <span><math><mrow><mi>τ</mi><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow><mo>≤</mo><mfrac><mrow><mn>2</mn></mrow><mrow><mn>7</mn></mrow></mfrac><mi>n</mi></mrow></math></span>. The Tuza constant <span><math><msub><mrow><mi>c</mi></mrow><mrow><mn>6</mn></mrow></msub></math></span> is defined by <span><math><mrow><msub><mrow><mi>c</mi></mrow><mrow><mn>6</mn></mrow></msub><mo>=</mo><mo>sup</mo><mfrac><mrow><mi>τ</mi><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow></mrow><mrow><mi>n</mi><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow><mo>+</mo><mi>m</mi><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow></mrow></mfrac></mrow></math></span>, where the supremum is taken over the class of all 6-uniform hypergraphs <span><math><mi>H</mi></math></span>. Since 1990 the exact value of <span><math><msub><mrow><mi>c</mi></mrow><mrow><mn>6</mn></mrow></msub></math></span> has yet to be determined. We show that <span><math><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>6</mn></mrow></mfrac><mo>≤</mo><msub><mrow><mi>c</mi></mrow><mrow><mn>6</mn></mrow></msub><mo>≤</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>6</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>210</mn></mrow></mfrac></mrow></math></span>, where <span><math><mrow><msub><mrow><mi>c</mi></mrow><mrow><mn>6</mn></mrow></msub><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>6</mn></mrow></mfrac></mrow></math></span> is conjectured to be the correct bound. Moreover we show that if <span><math><mi>G</mi></math></span> is a graph of order <span><math><mi>n</mi></math></span> with <span><math><mrow><mi>δ</mi><mrow><mo>(<
{"title":"Extensions and applications of the Tuza-Vestergaard theorem","authors":"Michael A. Henning , Anders Yeo","doi":"10.1016/j.ejc.2025.104201","DOIUrl":"10.1016/j.ejc.2025.104201","url":null,"abstract":"<div><div>The transversal number <span><math><mrow><mi>τ</mi><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow></mrow></math></span> of a hypergraph <span><math><mi>H</mi></math></span> is the minimum number of vertices that intersect every edge of <span><math><mi>H</mi></math></span>. A 6-uniform hypergraph has all edges of size 6. On 10 November 2000 Tuza and Vestergaard (2002) conjectured that if <span><math><mi>H</mi></math></span> is a 3-regular 6-uniform hypergraph of order <span><math><mi>n</mi></math></span>, then <span><math><mrow><mi>τ</mi><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow><mo>≤</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac><mi>n</mi></mrow></math></span>. This conjecture was recently proven by the Henning and Yeo (2023) and is now called the Tuza-Vestergaard Theorem. In this paper we extend the Tuza-Vestergaard Theorem by relaxing the 3-regularity constraint and allowing bounded maximum degree 4. We present several applications of the Tuza-Vestergaard Theorem and its extension. We obtain best known upper bounds to date on the transversal number of a (general) 6-uniform hypergraph <span><math><mi>H</mi></math></span> of order <span><math><mi>n</mi></math></span> and size <span><math><mi>m</mi></math></span>. In particular, if <span><math><mi>H</mi></math></span> is a 4-regular 6-uniform hypergraph of order <span><math><mi>n</mi></math></span>, then we show that <span><math><mrow><mi>τ</mi><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow><mo>≤</mo><mfrac><mrow><mn>2</mn></mrow><mrow><mn>7</mn></mrow></mfrac><mi>n</mi></mrow></math></span>. The Tuza constant <span><math><msub><mrow><mi>c</mi></mrow><mrow><mn>6</mn></mrow></msub></math></span> is defined by <span><math><mrow><msub><mrow><mi>c</mi></mrow><mrow><mn>6</mn></mrow></msub><mo>=</mo><mo>sup</mo><mfrac><mrow><mi>τ</mi><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow></mrow><mrow><mi>n</mi><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow><mo>+</mo><mi>m</mi><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow></mrow></mfrac></mrow></math></span>, where the supremum is taken over the class of all 6-uniform hypergraphs <span><math><mi>H</mi></math></span>. Since 1990 the exact value of <span><math><msub><mrow><mi>c</mi></mrow><mrow><mn>6</mn></mrow></msub></math></span> has yet to be determined. We show that <span><math><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>6</mn></mrow></mfrac><mo>≤</mo><msub><mrow><mi>c</mi></mrow><mrow><mn>6</mn></mrow></msub><mo>≤</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>6</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>210</mn></mrow></mfrac></mrow></math></span>, where <span><math><mrow><msub><mrow><mi>c</mi></mrow><mrow><mn>6</mn></mrow></msub><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>6</mn></mrow></mfrac></mrow></math></span> is conjectured to be the correct bound. Moreover we show that if <span><math><mi>G</mi></math></span> is a graph of order <span><math><mi>n</mi></math></span> with <span><math><mrow><mi>δ</mi><mrow><mo>(<","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"130 ","pages":"Article 104201"},"PeriodicalIF":1.0,"publicationDate":"2025-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144518192","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we propose a general framework that extends the theory of permutation patterns to higher dimensions and unifies several combinatorial objects studied in the literature. Our approach involves introducing the concept of a “level” for an element in a multi-dimensional permutation, which can be defined in multiple ways. We consider two natural definitions of a level, each establishing connections to other combinatorial sequences found in the Online Encyclopedia of Integer Sequences (OEIS).
Our framework allows us to offer combinatorial interpretations for various sequences found in the OEIS, many of which previously lacked such interpretations. As a notable example, we introduce an elegant combinatorial interpretation for the Springer numbers: they count weakly increasing 3-dimensional permutations under the definition of levels determined by maximal entries.
{"title":"Patterns in multi-dimensional permutations","authors":"Shaoshi Chen , Hanqian Fang , Sergey Kitaev , Candice X.T. Zhang","doi":"10.1016/j.ejc.2025.104203","DOIUrl":"10.1016/j.ejc.2025.104203","url":null,"abstract":"<div><div>In this paper, we propose a general framework that extends the theory of permutation patterns to higher dimensions and unifies several combinatorial objects studied in the literature. Our approach involves introducing the concept of a “level” for an element in a multi-dimensional permutation, which can be defined in multiple ways. We consider two natural definitions of a level, each establishing connections to other combinatorial sequences found in the Online Encyclopedia of Integer Sequences (OEIS).</div><div>Our framework allows us to offer combinatorial interpretations for various sequences found in the OEIS, many of which previously lacked such interpretations. As a notable example, we introduce an elegant combinatorial interpretation for the Springer numbers: they count weakly increasing 3-dimensional permutations under the definition of levels determined by maximal entries.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"130 ","pages":"Article 104203"},"PeriodicalIF":1.0,"publicationDate":"2025-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144518193","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-06-23DOI: 10.1016/j.ejc.2025.104199
Peter Frankl , Jian Wang
<div><div>Let <span><math><mrow><mi>F</mi><mo>⊂</mo><mfenced><mrow><mfrac><mrow><mi>X</mi></mrow><mrow><mi>k</mi></mrow></mfrac></mrow></mfenced></mrow></math></span> be a family consisting of <span><math><mi>k</mi></math></span>-subsets of the <span><math><mi>n</mi></math></span>-set <span><math><mi>X</mi></math></span>. Suppose that <span><math><mi>F</mi></math></span> is intersecting, i.e., <span><math><mrow><mi>F</mi><mo>∩</mo><msup><mrow><mi>F</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>≠</mo><mo>0̸</mo></mrow></math></span> for all <span><math><mrow><mi>F</mi><mo>,</mo><msup><mrow><mi>F</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>∈</mo><mi>F</mi></mrow></math></span>. Let <span><math><mrow><mi>Δ</mi><mrow><mo>(</mo><mi>F</mi><mo>)</mo></mrow></mrow></math></span> be the maximum degree of <span><math><mi>F</mi></math></span>. For a constant <span><math><mrow><mi>C</mi><mo>≥</mo><mn>1</mn></mrow></math></span> the <span><math><mi>C</mi></math></span><em>-diversity</em> <span><math><mrow><msub><mrow><mi>γ</mi></mrow><mrow><mi>C</mi></mrow></msub><mrow><mo>(</mo><mi>F</mi><mo>)</mo></mrow></mrow></math></span> is defined as <span><math><mrow><mrow><mo>|</mo><mi>F</mi><mo>|</mo></mrow><mo>−</mo><mi>C</mi><mi>Δ</mi><mrow><mo>(</mo><mi>F</mi><mo>)</mo></mrow></mrow></math></span>, which was introduced by Magnan, Palmer and Wood recently. Define <span><math><mrow><msub><mrow><mi>F</mi></mrow><mrow><mn>123</mn></mrow></msub><mo>=</mo><mfenced><mrow><mi>F</mi><mo>∈</mo><mfenced><mrow><mfrac><mrow><mi>X</mi></mrow><mrow><mi>k</mi></mrow></mfrac></mrow></mfenced><mo>:</mo><mrow><mo>|</mo><mi>F</mi><mo>∩</mo><mrow><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>}</mo></mrow><mo>|</mo></mrow><mo>=</mo><mn>2</mn></mrow></mfenced></mrow></math></span>. It has <span><math><mi>C</mi></math></span>-diversity <span><math><mrow><mrow><mo>(</mo><mn>3</mn><mo>−</mo><mn>2</mn><mi>C</mi><mo>)</mo></mrow><mfenced><mrow><mfrac><mrow><mi>n</mi><mo>−</mo><mn>3</mn></mrow><mrow><mi>k</mi><mo>−</mo><mn>2</mn></mrow></mfrac></mrow></mfenced></mrow></math></span>. The main result shows that for <span><math><mrow><mn>1</mn><mo><</mo><mi>C</mi><mo><</mo><mfrac><mrow><mn>3</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></math></span> and <span><math><mrow><mi>n</mi><mo>≥</mo><mfrac><mrow><mn>42</mn></mrow><mrow><mn>3</mn><mo>−</mo><mn>2</mn><mi>C</mi></mrow></mfrac><mi>k</mi></mrow></math></span>, <span><math><mrow><msub><mrow><mi>γ</mi></mrow><mrow><mi>C</mi></mrow></msub><mrow><mo>(</mo><mi>F</mi><mo>)</mo></mrow><mo>≤</mo><msub><mrow><mi>γ</mi></mrow><mrow><mi>C</mi></mrow></msub><mrow><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>123</mn></mrow></msub><mo>)</mo></mrow></mrow></math></span> with equality if and only if <span><math><mi>F</mi></math></span> is isomorphic to <span><math><msub><mrow><mi>F</mi></mrow><mrow><mn>123</mn></mrow></msub></math></span>. For the case of ordinary diversity <span><math><mrow><mo>(</mo><mi>C</mi><mo>=</mo><mn
{"title":"On the C-diversity of intersecting k-graphs","authors":"Peter Frankl , Jian Wang","doi":"10.1016/j.ejc.2025.104199","DOIUrl":"10.1016/j.ejc.2025.104199","url":null,"abstract":"<div><div>Let <span><math><mrow><mi>F</mi><mo>⊂</mo><mfenced><mrow><mfrac><mrow><mi>X</mi></mrow><mrow><mi>k</mi></mrow></mfrac></mrow></mfenced></mrow></math></span> be a family consisting of <span><math><mi>k</mi></math></span>-subsets of the <span><math><mi>n</mi></math></span>-set <span><math><mi>X</mi></math></span>. Suppose that <span><math><mi>F</mi></math></span> is intersecting, i.e., <span><math><mrow><mi>F</mi><mo>∩</mo><msup><mrow><mi>F</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>≠</mo><mo>0̸</mo></mrow></math></span> for all <span><math><mrow><mi>F</mi><mo>,</mo><msup><mrow><mi>F</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>∈</mo><mi>F</mi></mrow></math></span>. Let <span><math><mrow><mi>Δ</mi><mrow><mo>(</mo><mi>F</mi><mo>)</mo></mrow></mrow></math></span> be the maximum degree of <span><math><mi>F</mi></math></span>. For a constant <span><math><mrow><mi>C</mi><mo>≥</mo><mn>1</mn></mrow></math></span> the <span><math><mi>C</mi></math></span><em>-diversity</em> <span><math><mrow><msub><mrow><mi>γ</mi></mrow><mrow><mi>C</mi></mrow></msub><mrow><mo>(</mo><mi>F</mi><mo>)</mo></mrow></mrow></math></span> is defined as <span><math><mrow><mrow><mo>|</mo><mi>F</mi><mo>|</mo></mrow><mo>−</mo><mi>C</mi><mi>Δ</mi><mrow><mo>(</mo><mi>F</mi><mo>)</mo></mrow></mrow></math></span>, which was introduced by Magnan, Palmer and Wood recently. Define <span><math><mrow><msub><mrow><mi>F</mi></mrow><mrow><mn>123</mn></mrow></msub><mo>=</mo><mfenced><mrow><mi>F</mi><mo>∈</mo><mfenced><mrow><mfrac><mrow><mi>X</mi></mrow><mrow><mi>k</mi></mrow></mfrac></mrow></mfenced><mo>:</mo><mrow><mo>|</mo><mi>F</mi><mo>∩</mo><mrow><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>}</mo></mrow><mo>|</mo></mrow><mo>=</mo><mn>2</mn></mrow></mfenced></mrow></math></span>. It has <span><math><mi>C</mi></math></span>-diversity <span><math><mrow><mrow><mo>(</mo><mn>3</mn><mo>−</mo><mn>2</mn><mi>C</mi><mo>)</mo></mrow><mfenced><mrow><mfrac><mrow><mi>n</mi><mo>−</mo><mn>3</mn></mrow><mrow><mi>k</mi><mo>−</mo><mn>2</mn></mrow></mfrac></mrow></mfenced></mrow></math></span>. The main result shows that for <span><math><mrow><mn>1</mn><mo><</mo><mi>C</mi><mo><</mo><mfrac><mrow><mn>3</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></math></span> and <span><math><mrow><mi>n</mi><mo>≥</mo><mfrac><mrow><mn>42</mn></mrow><mrow><mn>3</mn><mo>−</mo><mn>2</mn><mi>C</mi></mrow></mfrac><mi>k</mi></mrow></math></span>, <span><math><mrow><msub><mrow><mi>γ</mi></mrow><mrow><mi>C</mi></mrow></msub><mrow><mo>(</mo><mi>F</mi><mo>)</mo></mrow><mo>≤</mo><msub><mrow><mi>γ</mi></mrow><mrow><mi>C</mi></mrow></msub><mrow><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>123</mn></mrow></msub><mo>)</mo></mrow></mrow></math></span> with equality if and only if <span><math><mi>F</mi></math></span> is isomorphic to <span><math><msub><mrow><mi>F</mi></mrow><mrow><mn>123</mn></mrow></msub></math></span>. For the case of ordinary diversity <span><math><mrow><mo>(</mo><mi>C</mi><mo>=</mo><mn","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"130 ","pages":"Article 104199"},"PeriodicalIF":1.0,"publicationDate":"2025-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144338491","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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