European Journal of Combinatorics最新文献

{"title":"A polynomial Ramsey statement for bounded VC-dimension","authors":"Tomáš Hons","doi":"10.1016/j.ejc.2025.104303","DOIUrl":"10.1016/j.ejc.2025.104303","url":null,"abstract":"<div><div>A theorem by Ding, Oporowski, Oxley, and Vertigan states that every sufficiently large bipartite graph without twins contains a matching, co-matching, or half-graph of any given size as an induced subgraph. We prove that this Ramsey statement has polynomial dependency assuming bounded VC-dimension of the initial graph, using the recent verification of the Erdős-Hajnal property for graphs of bounded VC-dimension. Since the theorem of Ding et al. plays a role in (finite) model theory, which studies even more restricted structures, we also comment on further refinements of the theorem within this context.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"133 ","pages":"Article 104303"},"PeriodicalIF":0.9,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145652036","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":"Maximizing the number of rational-value sums or zero-sums","authors":"Benjamin Móricz ,&nbsp;Zoltán Lóránt Nagy","doi":"10.1016/j.ejc.2025.104324","DOIUrl":"10.1016/j.ejc.2025.104324","url":null,"abstract":"<div><div>What is the maximum number of <span><math><mi>r</mi></math></span>-term sums admitting rational values in <span><math><mi>n</mi></math></span>-element sets of irrational numbers? We determine the maximum when <span><math><mrow><mi>r</mi><mo>&lt;</mo><mn>4</mn></mrow></math></span> or <span><math><mrow><mi>r</mi><mo>≥</mo><mi>n</mi><mo>/</mo><mn>2</mn></mrow></math></span> and also in case when we drop the condition on the number of summands. It turns out that the <span><math><mi>r</mi></math></span>-term sum problem is equivalent to determine the maximum number of <span><math><mi>r</mi></math></span>-term zero-sum subsequences in <span><math><mi>n</mi></math></span>-element sequences of integers, which can be seen as a variant of the famous Erdős–Ginzburg–Ziv theorem.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"133 ","pages":"Article 104324"},"PeriodicalIF":0.9,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145884522","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":"On the minimum spanning tree distribution in grids","authors":"Kristopher Tapp","doi":"10.1016/j.ejc.2025.104325","DOIUrl":"10.1016/j.ejc.2025.104325","url":null,"abstract":"<div><div>We study the minimum spanning tree distribution on the space of spanning trees of the <span><math><mi>n</mi></math></span>-by-<span><math><mi>n</mi></math></span> grid for large <span><math><mi>n</mi></math></span>. We establish bounds on the decay rates of the probability of the most and the least probable spanning trees as <span><math><mrow><mi>n</mi><mo>→</mo><mi>∞</mi></mrow></math></span>, and we develop general tools for studying the decay rates of spanning tree families.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"133 ","pages":"Article 104325"},"PeriodicalIF":0.9,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145840442","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":"On the separating Noether number of finite abelian groups","authors":"Barna Schefler ,&nbsp;Kevin Zhao ,&nbsp;Qinghai Zhong","doi":"10.1016/j.ejc.2025.104302","DOIUrl":"10.1016/j.ejc.2025.104302","url":null,"abstract":"<div><div>The separating Noether number <span><math><mrow><msub><mrow><mi>β</mi></mrow><mrow><mi>sep</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> of a finite group <span><math><mi>G</mi></math></span> is the minimal positive integer <span><math><mi>d</mi></math></span> such that for every finite dimensional <span><math><mi>G</mi></math></span>-module <span><math><mi>V</mi></math></span> there is a separating set consisting of invariant polynomials of degree at most <span><math><mi>d</mi></math></span>. In this paper we use methods from additive combinatorics to investigate the separating Noether number for finite abelian groups. Among others, we obtain the exact value of <span><math><mrow><msub><mrow><mi>β</mi></mrow><mrow><mi>sep</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, provided that <span><math><mi>G</mi></math></span> is either a <span><math><mi>p</mi></math></span>-group or has rank 2, 3 or 5.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"133 ","pages":"Article 104302"},"PeriodicalIF":0.9,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145652037","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":"Slit-slide-sew bijections for constellations and quasiconstellations","authors":"Jérémie Bettinelli ,&nbsp;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":"2026-03-01","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}

{"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":"2026-03-01","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}

{"title":"A new bijective proof of the q-Pfaff–Saalschütz identity with applications to quantum groups","authors":"Álvaro Gutiérrez ,&nbsp;Álvaro L. Martínez ,&nbsp;Michał Szwej ,&nbsp;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":"2026-03-01","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}

{"title":"Neighborly boxes and bipartite coverings; constructions and conjectures","authors":"Jarosław Grytczuk ,&nbsp;Andrzej P. Kisielewicz ,&nbsp;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":"2026-03-01","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}

{"title":"Restricted chain-order polytopes via combinatorial mutations","authors":"Oliver Clarke ,&nbsp;Akihiro Higashitani ,&nbsp;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":"2026-03-01","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":"Degree-truncated DP-colourability of K2,4-minor-free graphs","authors":"On-Hei Solomon Lo ,&nbsp;Cheng Wang ,&nbsp;Huan Zhou ,&nbsp;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":"2026-03-01","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}