European Journal of Combinatorics最新文献

{"title":"When all directed cycles have length three","authors":"Paul Seymour","doi":"10.1016/j.ejc.2025.104161","DOIUrl":"10.1016/j.ejc.2025.104161","url":null,"abstract":"<div><div>We give a construction to build all digraphs with the property that every directed cycle has length three.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"127 ","pages":"Article 104161"},"PeriodicalIF":1.0,"publicationDate":"2025-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143873957","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":"Grothendieck polynomials of inverse fireworks permutations","authors":"Chen-An (Jack) Chou ,&nbsp;Tianyi Yu","doi":"10.1016/j.ejc.2025.104158","DOIUrl":"10.1016/j.ejc.2025.104158","url":null,"abstract":"<div><div>Pipe dreams are combinatorial objects that compute Grothendieck polynomials. We introduce a new combinatorial object that naturally recasts the pipe dream formula. From this, we obtain the first direct combinatorial formula for the top degree components of Grothendieck polynomials, also known as the Castelnuovo–Mumford polynomials. We also prove the inverse fireworks case of a conjecture of Mészáros, Setiabrata, and St. Dizier on the support of Grothendieck polynomials.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"127 ","pages":"Article 104158"},"PeriodicalIF":1.0,"publicationDate":"2025-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143784054","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Distributions of parity differences and biases in partitions into distinct parts","authors":"Siu Hang Man","doi":"10.1016/j.ejc.2025.104157","DOIUrl":"10.1016/j.ejc.2025.104157","url":null,"abstract":"<div><div>For a partition <span><math><mrow><mi>λ</mi><mo>⊢</mo><mi>n</mi></mrow></math></span>, we let <span><math><mrow><mo>pd</mo><mrow><mo>(</mo><mi>λ</mi><mo>)</mo></mrow></mrow></math></span>, the parity difference of <span><math><mi>λ</mi></math></span>, be the number of odd parts of <span><math><mi>λ</mi></math></span> minus the number of even parts of <span><math><mi>λ</mi></math></span>. We prove for <span><math><mrow><msub><mrow><mi>c</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>∈</mo><mi>R</mi></mrow></math></span> an asymptotic expansion for the number of partitions of <span><math><mi>n</mi></math></span> into distinct parts with normalised parity difference <span><math><mrow><msup><mrow><mi>n</mi></mrow><mrow><mo>−</mo><mn>1</mn><mo>/</mo><mn>4</mn></mrow></msup><mo>pd</mo><mrow><mo>(</mo><mi>λ</mi><mo>)</mo></mrow></mrow></math></span> greater than <span><math><msub><mrow><mi>c</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> as <span><math><mrow><mi>n</mi><mo>→</mo><mi>∞</mi></mrow></math></span>. As a corollary, we find the distribution of the parity differences and parity biases for partitions of <span><math><mi>n</mi></math></span> into distinct parts. We also establish analogous results for generalised parity differences modulo <span><math><mi>N</mi></math></span>.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"127 ","pages":"Article 104157"},"PeriodicalIF":1.0,"publicationDate":"2025-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143759058","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":"Extremal problems for intersecting families of subspaces with a measure","authors":"Hajime Tanaka ,&nbsp;Norihide Tokushige","doi":"10.1016/j.ejc.2025.104156","DOIUrl":"10.1016/j.ejc.2025.104156","url":null,"abstract":"<div><div>We introduce a measure for subspaces of a vector space over a <span><math><mi>q</mi></math></span>-element field, and propose some extremal problems for intersecting families. These are <span><math><mi>q</mi></math></span>-analogue of Erdős–Ko–Rado type problems, and we answer some of the basic questions.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"127 ","pages":"Article 104156"},"PeriodicalIF":1.0,"publicationDate":"2025-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143714877","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 hypergraph Turán problems with bounded matching number","authors":"Dániel Gerbner ,&nbsp;Casey Tompkins ,&nbsp;Junpeng Zhou","doi":"10.1016/j.ejc.2025.104155","DOIUrl":"10.1016/j.ejc.2025.104155","url":null,"abstract":"<div><div>Very recently, Alon and Frankl, and Gerbner studied the maximum number of edges in <span><math><mi>n</mi></math></span>-vertex <span><math><mi>F</mi></math></span>-free graphs with bounded matching number, respectively. We consider the analogous Turán problems on hypergraphs with bounded matching number, and we obtain some exact results.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"127 ","pages":"Article 104155"},"PeriodicalIF":1.0,"publicationDate":"2025-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143704751","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The emergence of a giant rainbow component","authors":"Oliver Cooley ,&nbsp;Tuan Anh Do ,&nbsp;Joshua Erde ,&nbsp;Michael Missethan","doi":"10.1016/j.ejc.2025.104154","DOIUrl":"10.1016/j.ejc.2025.104154","url":null,"abstract":"<div><div>The <em>random coloured graph</em> <span><math><mrow><msub><mrow><mi>G</mi></mrow><mrow><mi>c</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>p</mi><mo>)</mo></mrow></mrow></math></span> is obtained from the Erdős–Rényi binomial random graph <span><math><mrow><mi>G</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>p</mi><mo>)</mo></mrow></mrow></math></span> by assigning to each edge a colour from a set of <span><math><mi>c</mi></math></span> colours independently and uniformly at random. It is not hard to see that, when <span><math><mrow><mi>c</mi><mo>=</mo><mi>Θ</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span>, the order of the largest rainbow tree in this model undergoes a phase transition at the critical point <span><math><mrow><mi>p</mi><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>n</mi></mrow></mfrac></mrow></math></span>. In this paper, we determine the asymptotic order of the largest rainbow tree in the <em>weakly sub- and supercritical regimes</em>, when <span><math><mrow><mi>p</mi><mo>=</mo><mfrac><mrow><mn>1</mn><mo>+</mo><mi>ɛ</mi></mrow><mrow><mi>n</mi></mrow></mfrac></mrow></math></span> for some <span><math><mrow><mi>ɛ</mi><mo>=</mo><mi>ɛ</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span> which satisfies <span><math><mrow><mi>ɛ</mi><mo>=</mo><mi>o</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><msup><mrow><mrow><mo>|</mo><mi>ɛ</mi><mo>|</mo></mrow></mrow><mrow><mn>3</mn></mrow></msup><mi>n</mi><mo>→</mo><mi>∞</mi></mrow></math></span>. In particular, we show that in both of these regimes with high probability the largest component of <span><math><mrow><msub><mrow><mi>G</mi></mrow><mrow><mi>c</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>p</mi><mo>)</mo></mrow></mrow></math></span> contains a rainbow tree which is almost spanning. We also consider the order of the largest rainbow tree in the <em>sparse regime</em>, when <span><math><mrow><mi>p</mi><mo>=</mo><mfrac><mrow><mi>d</mi></mrow><mrow><mi>n</mi></mrow></mfrac></mrow></math></span> for some constant <span><math><mrow><mi>d</mi><mo>&gt;</mo><mn>1</mn></mrow></math></span>. Here we show that the largest rainbow tree has linear order, and, moreover, for <span><math><mi>d</mi></math></span> and <span><math><mi>c</mi></math></span> sufficiently large, with high probability <span><math><mrow><msub><mrow><mi>G</mi></mrow><mrow><mi>c</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>p</mi><mo>)</mo></mrow></mrow></math></span> even contains a rainbow cycle which is almost spanning.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"127 ","pages":"Article 104154"},"PeriodicalIF":1.0,"publicationDate":"2025-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143696745","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Extensions of transversal valuated matroids","authors":"Alex Fink ,&nbsp;Jorge Alberto Olarte","doi":"10.1016/j.ejc.2025.104153","DOIUrl":"10.1016/j.ejc.2025.104153","url":null,"abstract":"<div><div>Following up on our previous work, we study single-element extensions of transversal valuated matroids. We show that tropical presentations of valuated matroids with a minimal set of finite entries enjoy counterparts of the properties proved by Bonin and de Mier of minimal non-valuated transversal presentations.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"127 ","pages":"Article 104153"},"PeriodicalIF":1.0,"publicationDate":"2025-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143643770","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Minors, connectivity, and diameter in randomly perturbed sparse graphs","authors":"Elad Aigner-Horev ,&nbsp;Dan Hefetz ,&nbsp;Michael Krivelevich","doi":"10.1016/j.ejc.2025.104152","DOIUrl":"10.1016/j.ejc.2025.104152","url":null,"abstract":"<div><div>Extremal properties of sparse graphs, randomly perturbed by the binomial random graph are considered. It is known that every <span><math><mi>n</mi></math></span>-vertex graph <span><math><mi>G</mi></math></span> contains a complete minor of order <span><math><mrow><mi>Ω</mi><mrow><mo>(</mo><mi>n</mi><mo>/</mo><mi>α</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></math></span>. We prove that adding <span><math><mrow><mi>ξ</mi><mi>n</mi></mrow></math></span> random edges, where <span><math><mrow><mi>ξ</mi><mo>&gt;</mo><mn>0</mn></mrow></math></span> is arbitrarily small yet fixed, to an <span><math><mi>n</mi></math></span>-vertex graph <span><math><mi>G</mi></math></span> satisfying <span><math><mrow><mi>α</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mi>ζ</mi><mrow><mo>(</mo><mi>ξ</mi><mo>)</mo></mrow><mi>n</mi></mrow></math></span> asymptotically almost surely results in a graph containing a complete minor of order <span><math><mrow><mover><mrow><mi>Ω</mi></mrow><mrow><mo>̃</mo></mrow></mover><mfenced><mrow><mi>n</mi><mo>/</mo><msqrt><mrow><mi>α</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></msqrt></mrow></mfenced></mrow></math></span>; this result is tight up to the implicit logarithmic terms.</div><div>For complete topological minors, we prove that there exists a constant <span><math><mrow><mi>C</mi><mo>&gt;</mo><mn>0</mn></mrow></math></span> such that adding <span><math><mrow><mi>C</mi><mi>n</mi></mrow></math></span> random edges to a graph <span><math><mi>G</mi></math></span> satisfying <span><math><mrow><mi>δ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mi>ω</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></math></span>, asymptotically almost surely results in a graph containing a complete topological minor of order <span><math><mrow><mover><mrow><mi>Ω</mi></mrow><mrow><mo>̃</mo></mrow></mover><mrow><mo>(</mo><mo>min</mo><mrow><mo>{</mo><mi>δ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>,</mo><msqrt><mrow><mi>n</mi></mrow></msqrt><mo>}</mo></mrow><mo>)</mo></mrow></mrow></math></span>; this result is tight up to the implicit logarithmic terms.</div><div>Finally, extending results of Bohman, Frieze, Krivelevich, and Martin for the dense case, we analyse the asymptotic behaviour of the vertex-connectivity and the diameter of randomly perturbed sparse graphs.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"127 ","pages":"Article 104152"},"PeriodicalIF":1.0,"publicationDate":"2025-03-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143631754","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":"Generating functions for fixed points of the Mullineux map","authors":"David J. Hemmer","doi":"10.1016/j.ejc.2025.104141","DOIUrl":"10.1016/j.ejc.2025.104141","url":null,"abstract":"<div><div>Mullineux defined an involution on the set of <span><math><mi>e</mi></math></span>-regular partitions of <span><math><mi>n</mi></math></span>. When <span><math><mrow><mi>e</mi><mo>=</mo><mi>p</mi></mrow></math></span> is prime, these partitions label irreducible symmetric group modules in characteristic <span><math><mi>p</mi></math></span>. Mullineux’s conjecture, since proven, was that this “Mullineux map” described the effect on the labels of taking the tensor product with the one-dimensional signature representation. Counting irreducible modules fixed by this tensor product is related to counting irreducible modules for the alternating group <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> in prime characteristic. In 1991, Andrews and Olsson worked out the generating function counting fixed points of Mullineux’s map when <span><math><mrow><mi>e</mi><mo>=</mo><mi>p</mi></mrow></math></span> is an odd prime (providing evidence in support of Mullineux’s conjecture). In 1998, Bessenrodt and Olsson counted the fixed points in a <span><math><mi>p</mi></math></span>-block of weight <span><math><mi>w</mi></math></span>. We extend both results to arbitrary <span><math><mi>e</mi></math></span>, and determine the corresponding generating functions. When <span><math><mi>e</mi></math></span> is odd but not prime the extension is immediate, while <span><math><mi>e</mi></math></span> even requires additional work and the results, which are different, have not appeared in the literature.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"127 ","pages":"Article 104141"},"PeriodicalIF":1.0,"publicationDate":"2025-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143629043","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 refinement on spectral Mantel’s theorem","authors":"Zhenzhen Lou ,&nbsp;Lu Lu ,&nbsp;Mingqing Zhai","doi":"10.1016/j.ejc.2025.104142","DOIUrl":"10.1016/j.ejc.2025.104142","url":null,"abstract":"<div><div>The well-known Mantel’s theorem states that <span><math><mrow><mi>e</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mrow><mo>⌊</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>/</mo><mn>4</mn><mo>⌋</mo></mrow></mrow></math></span> for every <span><math><mi>n</mi></math></span>-vertex triangle-free graph <span><math><mi>G</mi></math></span>. In 1970, Nosal showed a spectral version of Mantel’s theorem, which states that <span><math><mrow><mi>ρ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><msqrt><mrow><mi>m</mi></mrow></msqrt></mrow></math></span> for every triangle-free graph <span><math><mi>G</mi></math></span> on <span><math><mi>m</mi></math></span> edges. Later, Nikiforov proved that the equality holds in Nosal’s bound if and only if <span><math><mi>G</mi></math></span> is a complete bipartite graph. Lin, Ning, and Wu [Combin. Probab. Comput. 30 (2021)] first gave a result on non-bipartite triangle-free graphs. They proved that <span><math><mrow><mi>ρ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><msqrt><mrow><mi>m</mi><mo>−</mo><mn>1</mn></mrow></msqrt></mrow></math></span>, and equality holds if and only if <span><math><mi>G</mi></math></span> is a 5-cycle. Their result is actually stronger than Mantel’s theorem. Recently, Li, Feng and Peng [Electron. J. Combin. 31 (2024)] characterized the extremal non-bipartite triangle-free graphs with maximal spectral radius for even <span><math><mi>m</mi></math></span>. Furthermore, they proposed a question as follows: what is the extremal graph with maximal spectral radius over all non-bipartite <span><math><mrow><mo>{</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>,</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>5</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn><mi>ℓ</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>}</mo></mrow></math></span>-free graphs of even size <span><math><mi>m</mi></math></span>? This question is completely solved in this paper. Here we use a different technique, and we call it residual function, which can present more concise proofs on related problems. Finally, a general question is proposed for further research.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"127 ","pages":"Article 104142"},"PeriodicalIF":1.0,"publicationDate":"2025-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143629042","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}