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}
Pub Date : 2025-12-08DOI: 10.1016/j.ejc.2025.104304
Teemu Lundström , Leonardo Saud Maia Leite
In the last decade, the order polytope of the zigzag poset has been thoroughly studied. A related poset, called crown poset, obtained by adding an extra cover relation between the endpoints of an even zigzag poset, is not so well understood. In this paper, we study the order polytopes of crown posets. We provide explicit formulas for their -vectors. We provide recursive formulas for their Ehrhart polynomial, giving a counterpart to formulas found in the zigzag case by Petersen and Zhuang (2025). We use these formulas to simplify a computation by Ferroni, Morales and Panova (2025) of the linear term of the order polynomial of these posets. Furthermore, we provide a combinatorial interpretation for the coefficients of the -polynomial in terms of the cyclic swap statistic on cyclically alternating permutations, which provides a circular version of a result by Coons and Sullivant (2023).
{"title":"Order polytopes of crown posets","authors":"Teemu Lundström , Leonardo Saud Maia Leite","doi":"10.1016/j.ejc.2025.104304","DOIUrl":"10.1016/j.ejc.2025.104304","url":null,"abstract":"<div><div>In the last decade, the order polytope of the zigzag poset has been thoroughly studied. A related poset, called <em>crown poset</em>, obtained by adding an extra cover relation between the endpoints of an even zigzag poset, is not so well understood. In this paper, we study the order polytopes of crown posets. We provide explicit formulas for their <span><math><mi>f</mi></math></span>-vectors. We provide recursive formulas for their Ehrhart polynomial, giving a counterpart to formulas found in the zigzag case by Petersen and Zhuang (2025). We use these formulas to simplify a computation by Ferroni, Morales and Panova (2025) of the linear term of the order polynomial of these posets. Furthermore, we provide a combinatorial interpretation for the coefficients of the <span><math><msup><mrow><mi>h</mi></mrow><mrow><mo>∗</mo></mrow></msup></math></span>-polynomial in terms of the cyclic swap statistic on cyclically alternating permutations, which provides a circular version of a result by Coons and Sullivant (2023).</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"133 ","pages":"Article 104304"},"PeriodicalIF":0.9,"publicationDate":"2025-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145737871","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 : 2025-12-04DOI: 10.1016/j.ejc.2025.104298
Jonah Berggren , Jeremy L. Martin , José A. Samper
A matroid base polytope is a polytope in which each vertex has 0,1 coordinates and each edge is parallel to a difference of two coordinate vectors. Matroid base polytopes are described combinatorially by integral submodular functions on a boolean lattice, satisfying the unit increase property. We define a more general class of unbounded matroids, or U-matroids, by replacing the boolean lattice with an arbitrary distributive lattice. U-matroids thus serve as a combinatorial model for polyhedra that satisfy the vertex and edge conditions of matroid base polytopes, but may be unbounded. Like polymatroids, U-matroids generalize matroids and arise as a special case of submodular systems. We prove that every U-matroid admits a canonical largest extension to a matroid, which we call the generous extension; the analogous geometric statement is that every U-matroid base polyhedron contains a unique largest matroid base polytope. We show that the supports of vertices of a U-matroid base polyhedron span a shellable simplicial complex, and we characterize U-matroid basis systems in terms of shelling orders, generalizing Björner’s and Gale’s criteria for a simplicial complex to be a matroid independence complex. Finally, we present an application of our theory to subspace arrangements and show that the generous extension has a natural geometric interpretation in this setting.
{"title":"Unbounded matroids","authors":"Jonah Berggren , Jeremy L. Martin , José A. Samper","doi":"10.1016/j.ejc.2025.104298","DOIUrl":"10.1016/j.ejc.2025.104298","url":null,"abstract":"<div><div>A matroid base polytope is a polytope in which each vertex has 0,1 coordinates and each edge is parallel to a difference of two coordinate vectors. Matroid base polytopes are described combinatorially by integral submodular functions on a boolean lattice, satisfying the unit increase property. We define a more general class of <em>unbounded matroids</em>, or <em>U-matroids</em>, by replacing the boolean lattice with an arbitrary distributive lattice. U-matroids thus serve as a combinatorial model for polyhedra that satisfy the vertex and edge conditions of matroid base polytopes, but may be unbounded. Like polymatroids, U-matroids generalize matroids and arise as a special case of submodular systems. We prove that every U-matroid admits a canonical largest extension to a matroid, which we call the <em>generous extension</em>; the analogous geometric statement is that every U-matroid base polyhedron contains a unique largest matroid base polytope. We show that the supports of vertices of a U-matroid base polyhedron span a shellable simplicial complex, and we characterize U-matroid basis systems in terms of shelling orders, generalizing Björner’s and Gale’s criteria for a simplicial complex to be a matroid independence complex. Finally, we present an application of our theory to subspace arrangements and show that the generous extension has a natural geometric interpretation in this setting.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"133 ","pages":"Article 104298"},"PeriodicalIF":0.9,"publicationDate":"2025-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145685592","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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