Pub Date : 2025-12-22DOI: 10.1016/j.ejc.2025.104322
Huan Xiong, Lihong Yang
Simultaneous core partitions have been extensively studied over the past two decades. In 2013, Amdeberhan proposed several conjectures regarding the number, the average size, and the largest size of -core partitions with distinct parts. These conjectures were proved and generalized by Straub, Nath-Sellers, Zaleski-Zeilberger, Xiong, Paramonov, and many other mathematicians. In this paper, we introduce a natural self-conjugate partition analog of -core partitions with distinct parts and derive their number, average size, and largest size.
{"title":"A self-conjugate partition analog of (t,t+1)-core partitions with distinct parts","authors":"Huan Xiong, Lihong Yang","doi":"10.1016/j.ejc.2025.104322","DOIUrl":"10.1016/j.ejc.2025.104322","url":null,"abstract":"<div><div>Simultaneous core partitions have been extensively studied over the past two decades. In 2013, Amdeberhan proposed several conjectures regarding the number, the average size, and the largest size of <span><math><mrow><mo>(</mo><mi>t</mi><mo>,</mo><mi>t</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow></math></span>-core partitions with distinct parts. These conjectures were proved and generalized by Straub, Nath-Sellers, Zaleski-Zeilberger, Xiong, Paramonov, and many other mathematicians. In this paper, we introduce a natural self-conjugate partition analog of <span><math><mrow><mo>(</mo><mi>t</mi><mo>,</mo><mi>t</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow></math></span>-core partitions with distinct parts and derive their number, average size, and largest size.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"133 ","pages":"Article 104322"},"PeriodicalIF":0.9,"publicationDate":"2025-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145840445","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 : 2025-12-22DOI: 10.1016/j.ejc.2025.104321
Álvaro Gutiérrez , Álvaro L. Martínez , Michał Szwej , Mark Wildon
We present a combinatorial proof of the -Pfaff–Saalschütz identity by a composition of explicit bijections, in which -binomial coefficients are interpreted as counting subspaces of -vector spaces. As a corollary, we obtain a new multiplication rule for quantum binomial coefficients and hence a new presentation of Lusztig’s integral form of the Cartan subalgebra of the quantum group .
{"title":"A new bijective proof of the q-Pfaff–Saalschütz identity with applications to quantum groups","authors":"Álvaro Gutiérrez , Álvaro L. Martínez , Michał Szwej , Mark Wildon","doi":"10.1016/j.ejc.2025.104321","DOIUrl":"10.1016/j.ejc.2025.104321","url":null,"abstract":"<div><div>We present a combinatorial proof of the <span><math><mi>q</mi></math></span>-Pfaff–Saalschütz identity by a composition of explicit bijections, in which <span><math><mi>q</mi></math></span>-binomial coefficients are interpreted as counting subspaces of <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>-vector spaces. As a corollary, we obtain a new multiplication rule for quantum binomial coefficients and hence a new presentation of Lusztig’s integral form <span><math><mrow><msub><mrow><mi>U</mi></mrow><mrow><mi>Z</mi><mrow><mo>[</mo><mi>q</mi><mo>,</mo><msup><mrow><mi>q</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>]</mo></mrow></mrow></msub><mrow><mo>(</mo><msub><mrow><mi>sl</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mrow></math></span> of the Cartan subalgebra of the quantum group <span><math><mrow><msub><mrow><mi>U</mi></mrow><mrow><mi>q</mi></mrow></msub><mrow><mo>(</mo><msub><mrow><mi>sl</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></mrow></math></span>.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"133 ","pages":"Article 104321"},"PeriodicalIF":0.9,"publicationDate":"2025-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145840436","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 : 2025-12-22DOI: 10.1016/j.ejc.2025.104326
Oliver Clarke , Akihiro Higashitani , Francesca Zaffalon
We study restricted chain-order polytopes associated to Young diagrams using combinatorial mutations. These polytopes are obtained by intersecting chain-order polytopes with certain hyperplanes. The family of chain-order polytopes associated to a poset interpolate between the order and chain polytopes of the poset. Each such polytope retains properties of the order and chain polytope; for example its Ehrhart polynomial. For a fixed Young diagram, we show that all restricted chain-order polytopes are related by a sequence of combinatorial mutations. Since the property of giving rise to the period collapse phenomenon is invariant under combinatorial mutations, we provide a large class of rational polytopes that give rise to period collapse.
{"title":"Restricted chain-order polytopes via combinatorial mutations","authors":"Oliver Clarke , Akihiro Higashitani , Francesca Zaffalon","doi":"10.1016/j.ejc.2025.104326","DOIUrl":"10.1016/j.ejc.2025.104326","url":null,"abstract":"<div><div>We study restricted chain-order polytopes associated to Young diagrams using combinatorial mutations. These polytopes are obtained by intersecting chain-order polytopes with certain hyperplanes. The family of chain-order polytopes associated to a poset interpolate between the order and chain polytopes of the poset. Each such polytope retains properties of the order and chain polytope; for example its Ehrhart polynomial. For a fixed Young diagram, we show that all restricted chain-order polytopes are related by a sequence of combinatorial mutations. Since the property of giving rise to the period collapse phenomenon is invariant under combinatorial mutations, we provide a large class of rational polytopes that give rise to period collapse.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"133 ","pages":"Article 104326"},"PeriodicalIF":0.9,"publicationDate":"2025-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145840444","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}
{"title":"A characterization of the Grassmann graphs: One missing case","authors":"Jack H. Koolen , Chenhui Lv , Alexander L. Gavrilyuk","doi":"10.1016/j.ejc.2025.104320","DOIUrl":"10.1016/j.ejc.2025.104320","url":null,"abstract":"<div><div>We prove that the Grassmann graphs <span><math><mrow><msub><mrow><mi>J</mi></mrow><mrow><mn>2</mn></mrow></msub><mrow><mo>(</mo><mn>2</mn><mi>D</mi><mo>+</mo><mn>3</mn><mo>,</mo><mi>D</mi><mo>)</mo></mrow></mrow></math></span>, <span><math><mrow><mi>D</mi><mo>≥</mo><mn>3</mn></mrow></math></span>, are characterized by their intersection numbers, which settles one of the few remaining cases.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"133 ","pages":"Article 104320"},"PeriodicalIF":0.9,"publicationDate":"2025-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145791235","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 : 2025-12-20DOI: 10.1016/j.ejc.2025.104307
Sam Spiro
Given a digraph , we say that a set of vertices is a -kernel if is an independent set and if every vertex of can be reached from by a path of length at most . In this paper, we initiate the study of several extremal problems for -kernels. For example, we introduce and make progress on (what turns out to be) a weak version of the Small Quasikernel Conjecture, namely that every digraph contains a -kernel with for all .
{"title":"Generalized quasikernels in digraphs","authors":"Sam Spiro","doi":"10.1016/j.ejc.2025.104307","DOIUrl":"10.1016/j.ejc.2025.104307","url":null,"abstract":"<div><div>Given a digraph <span><math><mi>D</mi></math></span>, we say that a set of vertices <span><math><mrow><mi>Q</mi><mo>⊆</mo><mi>V</mi><mrow><mo>(</mo><mi>D</mi><mo>)</mo></mrow></mrow></math></span> is a <span><math><mi>q</mi></math></span>-kernel if <span><math><mi>Q</mi></math></span> is an independent set and if every vertex of <span><math><mi>D</mi></math></span> can be reached from <span><math><mi>Q</mi></math></span> by a path of length at most <span><math><mi>q</mi></math></span>. In this paper, we initiate the study of several extremal problems for <span><math><mi>q</mi></math></span>-kernels. For example, we introduce and make progress on (what turns out to be) a weak version of the Small Quasikernel Conjecture, namely that every digraph contains a <span><math><mi>q</mi></math></span>-kernel with <span><math><mrow><mrow><mo>|</mo><msup><mrow><mi>N</mi></mrow><mrow><mo>+</mo></mrow></msup><mrow><mo>[</mo><mi>Q</mi><mo>]</mo></mrow><mo>|</mo></mrow><mo>≥</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mrow><mo>|</mo><mi>V</mi><mrow><mo>(</mo><mi>D</mi><mo>)</mo></mrow><mo>|</mo></mrow></mrow></math></span> for all <span><math><mrow><mi>q</mi><mo>≥</mo><mn>2</mn></mrow></math></span>.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"133 ","pages":"Article 104307"},"PeriodicalIF":0.9,"publicationDate":"2025-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145791236","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 : 2025-12-19DOI: 10.1016/j.ejc.2025.104318
Jérémie Bettinelli , Dimitri Korkotashvili
We extend so-called slit-slide-sew bijections to constellations and quasiconstellations, which allow to recover the counting formula for constellations or quasiconstellations with a given face degree distribution.
More precisely, we present an involution on the set of hypermaps given with an orientation, one distinguished corner, and one distinguished edge leading away from the corner while oriented in the given orientation. This involution reverts the orientation, exchanges the distinguished corner with the distinguished edge in some sense, slightly modifying the degrees of the incident faces in passing, while keeping all the other faces intact.
The construction consists in building a canonical path from the distinguished elements, slitting the map along it, and sewing back after sliding by one unit along the path. The involution specializes into a bijection interpreting combinatorial identities linking the numbers of constellations or quasiconstellations with a given face degree distribution, where the degree distributions differ by one and one .
Our bijections yield a “degree-by-degree, face-by-face” growth algorithm that samples a hypermap uniformly distributed among constellations or quasiconstellations with prescribed face degrees. More precisely, it samples at each step uniform constellations or quasiconstellations, whose face degree distributions slightly evolve to the desired distribution.
{"title":"Slit-slide-sew bijections for constellations and quasiconstellations","authors":"Jérémie Bettinelli , Dimitri Korkotashvili","doi":"10.1016/j.ejc.2025.104318","DOIUrl":"10.1016/j.ejc.2025.104318","url":null,"abstract":"<div><div>We extend so-called slit-slide-sew bijections to constellations and quasiconstellations, which allow to recover the counting formula for constellations or quasiconstellations with a given face degree distribution.</div><div>More precisely, we present an involution on the set of hypermaps given with an orientation, one distinguished corner, and one distinguished edge leading away from the corner while oriented in the given orientation. This involution reverts the orientation, exchanges the distinguished corner with the distinguished edge in some sense, slightly modifying the degrees of the incident faces in passing, while keeping all the other faces intact.</div><div>The construction consists in building a canonical path from the distinguished elements, slitting the map along it, and sewing back after sliding by one unit along the path. The involution specializes into a bijection interpreting combinatorial identities linking the numbers of constellations or quasiconstellations with a given face degree distribution, where the degree distributions differ by one <span><math><mrow><mo>+</mo><mn>1</mn></mrow></math></span> and one <span><math><mrow><mo>−</mo><mn>1</mn></mrow></math></span>.</div><div>Our bijections yield a “degree-by-degree, face-by-face” growth algorithm that samples a hypermap uniformly distributed among constellations or quasiconstellations with prescribed face degrees. More precisely, it samples at each step uniform constellations or quasiconstellations, whose face degree distributions slightly evolve to the desired distribution.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"133 ","pages":"Article 104318"},"PeriodicalIF":0.9,"publicationDate":"2025-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145791234","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 : 2025-12-18DOI: 10.1016/j.ejc.2025.104319
Jarosław Grytczuk , Andrzej P. Kisielewicz , Krzysztof Przesławski
Two axis-aligned boxes in are -neighborly if their intersection has dimension at least and at most . The maximum number of pairwise -neighborly boxes in is denoted by . It is known that , for fixed , but exact formulas are known only in three cases: , , and . In particular, the formula is equivalent to the famous theorem of Graham and Pollak on bipartite partitions of cliques.
In this paper we are dealing with the case . We give a new construction of -neighborly codes giving better lower bounds on . The construction is recursive in nature and uses a kind of “algebra” on lists of ternary strings, which encode neighborly boxes in a familiar way. Moreover, we conjecture that our construction is optimal and gives an explicit formula for . This supposition is supported by some numerical experiments and some partial results on related open problems which are recalled.
{"title":"Neighborly boxes and bipartite coverings; constructions and conjectures","authors":"Jarosław Grytczuk , Andrzej P. Kisielewicz , Krzysztof Przesławski","doi":"10.1016/j.ejc.2025.104319","DOIUrl":"10.1016/j.ejc.2025.104319","url":null,"abstract":"<div><div>Two axis-aligned boxes in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> are <span><math><mi>k</mi></math></span>-<em>neighborly</em> if their intersection has dimension at least <span><math><mrow><mi>d</mi><mo>−</mo><mi>k</mi></mrow></math></span> and at most <span><math><mrow><mi>d</mi><mo>−</mo><mn>1</mn></mrow></math></span>. The maximum number of pairwise <span><math><mi>k</mi></math></span>-neighborly boxes in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> is denoted by <span><math><mrow><mi>n</mi><mrow><mo>(</mo><mi>k</mi><mo>,</mo><mi>d</mi><mo>)</mo></mrow></mrow></math></span>. It is known that <span><math><mrow><mi>n</mi><mrow><mo>(</mo><mi>k</mi><mo>,</mo><mi>d</mi><mo>)</mo></mrow><mo>=</mo><mi>Θ</mi><mrow><mo>(</mo><msup><mrow><mi>d</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span>, for fixed <span><math><mrow><mn>1</mn><mo>⩽</mo><mi>k</mi><mo>⩽</mo><mi>d</mi></mrow></math></span>, but exact formulas are known only in three cases: <span><math><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow></math></span>, <span><math><mrow><mi>k</mi><mo>=</mo><mi>d</mi><mo>−</mo><mn>1</mn></mrow></math></span>, and <span><math><mrow><mi>k</mi><mo>=</mo><mi>d</mi></mrow></math></span>. In particular, the formula <span><math><mrow><mi>n</mi><mrow><mo>(</mo><mn>1</mn><mo>,</mo><mi>d</mi><mo>)</mo></mrow><mo>=</mo><mi>d</mi><mo>+</mo><mn>1</mn></mrow></math></span> is equivalent to the famous theorem of Graham and Pollak on bipartite partitions of cliques.</div><div>In this paper we are dealing with the case <span><math><mrow><mi>k</mi><mo>=</mo><mn>2</mn></mrow></math></span>. We give a new construction of <span><math><mi>k</mi></math></span>-neighborly <em>codes</em> giving better lower bounds on <span><math><mrow><mi>n</mi><mrow><mo>(</mo><mn>2</mn><mo>,</mo><mi>d</mi><mo>)</mo></mrow></mrow></math></span>. The construction is recursive in nature and uses a kind of “algebra” on <em>lists</em> of ternary strings, which encode neighborly boxes in a familiar way. Moreover, we conjecture that our construction is optimal and gives an explicit formula for <span><math><mrow><mi>n</mi><mrow><mo>(</mo><mn>2</mn><mo>,</mo><mi>d</mi><mo>)</mo></mrow></mrow></math></span>. This supposition is supported by some numerical experiments and some partial results on related open problems which are recalled.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"133 ","pages":"Article 104319"},"PeriodicalIF":0.9,"publicationDate":"2025-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145791233","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 : 2025-12-16DOI: 10.1016/j.ejc.2025.104316
Ronen Wdowinski
Given a multi-hypergraph that is edge-colored into color classes , a full rainbow matching is a matching of that contains exactly one edge from each color class . One way to guarantee the existence of a full rainbow matching is to have the size of each color class be sufficiently large compared to the maximum degree of . In this paper, we apply an iterative method to construct edge-colored multi-hypergraphs with a given maximum degree, large color classes, and no full rainbow matchings. First, for every and , we construct edge-colored -uniform multi-hypergraphs with maximum degree such that each color class has size and there is no full rainbow matching, which demonstrates that a theorem of Aharoni, Berger, and Meshulam (2005) is best possible. Second, we construct properly edge-colored multigraphs with no full rainbow matchings which disprove conjectures of Delcourt and Postle (2022). Finally, we apply results on full rainbow matchings to list edge-colorings and prove that a color degree generalization of Galvin’s theorem (1995) does not hold.
给定一个多超图G，它的边被颜色划分为E1，…，En，那么全彩虹匹配就是G的匹配，它只包含来自每个颜色类Ei的一条边。保证完全彩虹匹配存在的一种方法是使每个颜色类Ei的大小相对于g的最大度足够大。在本文中，我们应用迭代方法构造具有给定最大度、大颜色类和无完全彩虹匹配的边缘彩色多超图。首先，对于r≥1和Δ≥2，我们构造了最大度为Δ的边色r-均匀多超图，使得每个颜色类的大小为|Ei|≥rΔ−1，并且不存在完全彩虹匹配，这证明了Aharoni， Berger, and Meshulam（2005）的定理是最好的。其次，我们构建了正确的边缘彩色多图，没有完整的彩虹匹配，这反驳了Delcourt和Postle（2022）的猜想。最后，我们应用全彩虹匹配的结果来列出边缘着色，并证明了Galvin定理（1995）的色度推广不成立。
{"title":"Bounded degree graphs and hypergraphs with no full rainbow matchings","authors":"Ronen Wdowinski","doi":"10.1016/j.ejc.2025.104316","DOIUrl":"10.1016/j.ejc.2025.104316","url":null,"abstract":"<div><div>Given a multi-hypergraph <span><math><mi>G</mi></math></span> that is edge-colored into color classes <span><math><mrow><msub><mrow><mi>E</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></math></span>, a full rainbow matching is a matching of <span><math><mi>G</mi></math></span> that contains exactly one edge from each color class <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span>. One way to guarantee the existence of a full rainbow matching is to have the size of each color class <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> be sufficiently large compared to the maximum degree of <span><math><mi>G</mi></math></span>. In this paper, we apply an iterative method to construct edge-colored multi-hypergraphs with a given maximum degree, large color classes, and no full rainbow matchings. First, for every <span><math><mrow><mi>r</mi><mo>≥</mo><mn>1</mn></mrow></math></span> and <span><math><mrow><mi>Δ</mi><mo>≥</mo><mn>2</mn></mrow></math></span>, we construct edge-colored <span><math><mi>r</mi></math></span>-uniform multi-hypergraphs with maximum degree <span><math><mi>Δ</mi></math></span> such that each color class has size <span><math><mrow><mrow><mo>|</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>|</mo></mrow><mo>≥</mo><mi>r</mi><mi>Δ</mi><mo>−</mo><mn>1</mn></mrow></math></span> and there is no full rainbow matching, which demonstrates that a theorem of Aharoni, Berger, and Meshulam (2005) is best possible. Second, we construct properly edge-colored multigraphs with no full rainbow matchings which disprove conjectures of Delcourt and Postle (2022). Finally, we apply results on full rainbow matchings to list edge-colorings and prove that a color degree generalization of Galvin’s theorem (1995) does not hold.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"133 ","pages":"Article 104316"},"PeriodicalIF":0.9,"publicationDate":"2025-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145791231","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 : 2025-12-16DOI: 10.1016/j.ejc.2025.104317
Itai Benjamini, Or Bernard Kalifa, Elad Tzalik
A graph is -minor-universal if every graph with at most edges (and no isolated vertices) is a minor of . We prove that the -dimensional hypercube, , is -minor-universal, and that there exists an absolute constant such that is not -minor-universal. Similar results are obtained in a more general setting, where we bound the size of minors in a product of finite connected graphs. A key component of our proof is the following claim regarding the decomposition of a permutation of a box into simpler, one-dimensional permutations: Let be positive integers, and define . We prove that every permutation can be expressed as , where each is a one-dimensional permutation, meaning it fixes all coordinates except possibly one. We discuss future directions and pose open problems.
{"title":"Hypercube minor-universality","authors":"Itai Benjamini, Or Bernard Kalifa, Elad Tzalik","doi":"10.1016/j.ejc.2025.104317","DOIUrl":"10.1016/j.ejc.2025.104317","url":null,"abstract":"<div><div>A graph <span><math><mi>G</mi></math></span> is <span><math><mi>m</mi></math></span>-minor-universal if every graph with at most <span><math><mi>m</mi></math></span> edges (and no isolated vertices) is a minor of <span><math><mi>G</mi></math></span>. We prove that the <span><math><mi>d</mi></math></span>-dimensional hypercube, <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>d</mi></mrow></msub></math></span>, is <span><math><mrow><mi>Ω</mi><mfenced><mrow><mfrac><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>d</mi></mrow></msup></mrow><mrow><mi>d</mi></mrow></mfrac></mrow></mfenced></mrow></math></span>-minor-universal, and that there exists an absolute constant <span><math><mrow><mi>C</mi><mo>></mo><mn>0</mn></mrow></math></span> such that <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>d</mi></mrow></msub></math></span> is not <span><math><mfrac><mrow><mi>C</mi><msup><mrow><mn>2</mn></mrow><mrow><mi>d</mi></mrow></msup></mrow><mrow><msqrt><mrow><mi>d</mi></mrow></msqrt></mrow></mfrac></math></span>-minor-universal. Similar results are obtained in a more general setting, where we bound the size of minors in a product of finite connected graphs. A key component of our proof is the following claim regarding the decomposition of a permutation of a box into simpler, one-dimensional permutations: Let <span><math><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>n</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></math></span> be positive integers, and define <span><math><mrow><mi>X</mi><mo>≔</mo><mrow><mo>[</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>]</mo></mrow><mo>×</mo><mo>⋯</mo><mo>×</mo><mrow><mo>[</mo><msub><mrow><mi>n</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>]</mo></mrow></mrow></math></span>. We prove that every permutation <span><math><mrow><mi>σ</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>X</mi></mrow></math></span> can be expressed as <span><math><mrow><mi>σ</mi><mo>=</mo><msub><mrow><mi>σ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>∘</mo><mo>⋯</mo><mo>∘</mo><msub><mrow><mi>σ</mi></mrow><mrow><mn>2</mn><mi>d</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow></math></span>, where each <span><math><msub><mrow><mi>σ</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> is a <em>one-dimensional</em> permutation, meaning it fixes all coordinates except possibly one. We discuss future directions and pose open problems.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"133 ","pages":"Article 104317"},"PeriodicalIF":0.9,"publicationDate":"2025-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145791232","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 : 2025-12-12DOI: 10.1016/j.ejc.2025.104306
On-Hei Solomon Lo , Cheng Wang , Huan Zhou , Xuding Zhu
Assume is a graph and is a positive integer. Let be defined as . If is DP--colourable (respectively, -choosable), then we say is degree-truncated DP--colourable (respectively, degree-truncated -choosable). Hutchinson (2008) proved that 2-connected maximal outerplanar graphs other than the triangle are degree-truncated 5-choosable. Hutchinson asked whether the result can be extended to all outerplanar graphs. This paper proves that 2-connected -minor-free graphs other than cycles and complete graphs are degree-truncated DP-5-colourable. This not only answers Hutchinson’s question in the affirmative, but also extends to a larger family of graphs, and strengthens choosability to DP-colourability.
{"title":"Degree-truncated DP-colourability of K2,4-minor-free graphs","authors":"On-Hei Solomon Lo , Cheng Wang , Huan Zhou , Xuding Zhu","doi":"10.1016/j.ejc.2025.104306","DOIUrl":"10.1016/j.ejc.2025.104306","url":null,"abstract":"<div><div>Assume <span><math><mi>G</mi></math></span> is a graph and <span><math><mi>k</mi></math></span> is a positive integer. Let <span><math><mrow><mi>f</mi><mo>:</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>→</mo><mi>N</mi></mrow></math></span> be defined as <span><math><mrow><mi>f</mi><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow><mo>=</mo><mo>min</mo><mrow><mo>{</mo><mi>k</mi><mo>,</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>G</mi></mrow></msub><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow><mo>}</mo></mrow></mrow></math></span>. If <span><math><mi>G</mi></math></span> is DP-<span><math><mi>f</mi></math></span>-colourable (respectively, <span><math><mi>f</mi></math></span>-choosable), then we say <span><math><mi>G</mi></math></span> is degree-truncated DP-<span><math><mi>k</mi></math></span>-colourable (respectively, degree-truncated <span><math><mi>k</mi></math></span>-choosable). Hutchinson (2008) proved that 2-connected maximal outerplanar graphs other than the triangle are degree-truncated 5-choosable. Hutchinson asked whether the result can be extended to all outerplanar graphs. This paper proves that 2-connected <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn><mo>,</mo><mn>4</mn></mrow></msub></math></span>-minor-free graphs other than cycles and complete graphs are degree-truncated DP-5-colourable. This not only answers Hutchinson’s question in the affirmative, but also extends to a larger family of graphs, and strengthens choosability to DP-colourability.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"133 ","pages":"Article 104306"},"PeriodicalIF":0.9,"publicationDate":"2025-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145737872","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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