Pub Date : 2026-04-01Epub Date: 2026-01-30DOI: 10.1016/j.ejc.2026.104345
Théo Lenoir
We consider large uniform labeled random graphs in different classes with few induced ( is the graph consisting of a single line of 4 vertices) which generalize the case of cographs. Our main result is the convergence to a Brownian limit object in the space of graphons. As a by-product we obtain new asymptotic enumerative results for all these graph classes. We also obtain typical density results for a wide variety of labeled induced subgraphs. These asymptotics hold at a smaller scale than what is observable through the graphon convergence.
Our proofs rely on tree encoding of graphs. We then use mainly combinatorial arguments, including the symbolic method and singularity analysis.
{"title":"Graph classes with few P4’s: Universality and Brownian graphon limits","authors":"Théo Lenoir","doi":"10.1016/j.ejc.2026.104345","DOIUrl":"10.1016/j.ejc.2026.104345","url":null,"abstract":"<div><div>We consider large uniform labeled random graphs in different classes with few induced <span><math><msub><mrow><mi>P</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span>\u0000 (<span><math><msub><mrow><mi>P</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span> is the graph consisting of a single line of 4 vertices) which generalize the case of cographs. Our main result is the convergence to a Brownian limit object in the space of graphons. As a by-product we obtain new asymptotic enumerative results for all these graph classes. We also obtain typical density results for a wide variety of labeled induced subgraphs. These asymptotics hold at a smaller scale than what is observable through the graphon convergence.</div><div>Our proofs rely on tree encoding of graphs. We then use mainly combinatorial arguments, including the symbolic method and singularity analysis.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"134 ","pages":"Article 104345"},"PeriodicalIF":0.9,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146078666","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 : 2026-04-01Epub Date: 2026-01-05DOI: 10.1016/j.ejc.2025.104328
Ting Su
Matroids over skew tracts provide an algebraic framework simultaneously generalizing the notions of linear subspaces, matroids, oriented matroids, phased matroids, and some other “matroids with extra structure”. A single-element extension of a matroid over a skew tract is a matroid over obtained from by adding one more element. Crapo characterized single-element extensions of ordinary matroids, and Las Vergnas characterized single-element extensions of oriented matroids, in terms of single-element extensions of their rank 2 contractions. The results of Crapo and Las Vergnas do not generalize to matroids over skew tracts, but we will show a necessary and sufficient condition on skew tracts, called Pathetic Cancellation, such that the result can generalize to weak matroids over skew tracts.
Stringent skew hyperfields are a special case of skew tracts which behave in many ways like skew fields. We find a characterization of single-element extensions of strong matroids over stringent skew hyperfields.
{"title":"Single-element extensions of matroids over skew tracts","authors":"Ting Su","doi":"10.1016/j.ejc.2025.104328","DOIUrl":"10.1016/j.ejc.2025.104328","url":null,"abstract":"<div><div>Matroids over skew tracts provide an algebraic framework simultaneously generalizing the notions of linear subspaces, matroids, oriented matroids, phased matroids, and some other “matroids with extra structure”. A single-element extension of a matroid <span><math><mi>M</mi></math></span> over a skew tract <span><math><mi>T</mi></math></span> is a matroid <span><math><mover><mrow><mi>M</mi></mrow><mrow><mo>˜</mo></mrow></mover></math></span> over <span><math><mi>T</mi></math></span> obtained from <span><math><mi>M</mi></math></span> by adding one more element. Crapo characterized single-element extensions of ordinary matroids, and Las Vergnas characterized single-element extensions of oriented matroids, in terms of single-element extensions of their rank 2 contractions. The results of Crapo and Las Vergnas do not generalize to matroids over skew tracts, but we will show a necessary and sufficient condition on skew tracts, called Pathetic Cancellation, such that the result can generalize to weak matroids over skew tracts.</div><div>Stringent skew hyperfields are a special case of skew tracts which behave in many ways like skew fields. We find a characterization of single-element extensions of strong matroids over stringent skew hyperfields.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"134 ","pages":"Article 104328"},"PeriodicalIF":0.9,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145927941","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 : 2026-04-01Epub Date: 2025-12-31DOI: 10.1016/j.ejc.2025.104331
Jian Wang , Jimeng Xiao
Let be the symmetric group on the set . A family is called intersecting if for every there exists some such that . Deza and Frankl proved that the largest intersecting family of permutations is the full star, that is, the collection of all permutations with a fixed position. The diversity of an intersecting family is defined as the minimum number of permutations in , whose deletion results in a star. In the present paper, by applying the spread approximation method developed recently by Kupavskii and Zakharov, we prove that for the diversity of an intersecting subfamily of is at most , which is best possible.
{"title":"A note on the maximum diversity of intersecting families in the symmetric group","authors":"Jian Wang , Jimeng Xiao","doi":"10.1016/j.ejc.2025.104331","DOIUrl":"10.1016/j.ejc.2025.104331","url":null,"abstract":"<div><div>Let <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> be the symmetric group on the set <span><math><mrow><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow><mo>≔</mo><mrow><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo>}</mo></mrow></mrow></math></span>. A family <span><math><mrow><mi>F</mi><mo>⊂</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></math></span> is called intersecting if for every <span><math><mrow><mi>σ</mi><mo>,</mo><mi>π</mi><mo>∈</mo><mi>F</mi></mrow></math></span> there exists some <span><math><mrow><mi>i</mi><mo>∈</mo><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow></mrow></math></span> such that <span><math><mrow><mi>σ</mi><mrow><mo>(</mo><mi>i</mi><mo>)</mo></mrow><mo>=</mo><mi>π</mi><mrow><mo>(</mo><mi>i</mi><mo>)</mo></mrow></mrow></math></span>. Deza and Frankl proved that the largest intersecting family of permutations is the full star, that is, the collection of all permutations with a fixed position. The diversity of an intersecting family <span><math><mi>F</mi></math></span> is defined as the minimum number of permutations in <span><math><mi>F</mi></math></span>, whose deletion results in a star. In the present paper, by applying the spread approximation method developed recently by Kupavskii and Zakharov, we prove that for <span><math><mrow><mi>n</mi><mo>≥</mo><mn>500</mn></mrow></math></span> the diversity of an intersecting subfamily of <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> is at most <span><math><mrow><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mn>3</mn><mo>)</mo></mrow><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mn>3</mn><mo>)</mo></mrow><mo>!</mo></mrow></math></span>, which is best possible.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"134 ","pages":"Article 104331"},"PeriodicalIF":0.9,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145872451","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 : 2026-04-01Epub Date: 2026-01-19DOI: 10.1016/j.ejc.2025.104330
Andrzej Grzesik , Daniel Král , Samuel Mohr
The notion of first order convergence of graphs unifies the notions of convergence for sparse and dense graphs. Nešetřil and Ossona de Mendez (2019) proved that every first order convergent sequence of graphs from a nowhere-dense class of graphs has a modeling limit and conjectured the existence of such modeling limits with an additional property, the strong finitary mass transport principle. The existence of modeling limits satisfying the strong finitary mass transport principle was proved for first order convergent sequences of trees by Nešetřil and Ossona de Mendez (2016) and for first order sequences of graphs with bounded path-width by Gajarský et al. (2017). We establish the existence of modeling limits satisfying the strong finitary mass transport principle for first order convergent sequences of graphs with bounded tree-width.
图的一阶收敛性的概念统一了稀疏图和密集图的收敛性的概念。Nešetřil和Ossona de Mendez（2019）证明了来自无密度图类的每一个一阶收敛图序列都有一个建模极限，并通过一个附加性质，即强有限质量输运原理，推测了这种建模极限的存在。Nešetřil和Ossona de Mendez（2016）证明了树的一阶收敛序列满足强有限质量输运原理的建模极限的存在性，Gajarský等人（2017）证明了路径宽度有界的图的一阶序列满足强有限质量输运原理。建立了树宽有界的一阶收敛图序列满足强有限质量输运原理的建模极限的存在性。
{"title":"Strong modeling limits of graphs with bounded tree-width","authors":"Andrzej Grzesik , Daniel Král , Samuel Mohr","doi":"10.1016/j.ejc.2025.104330","DOIUrl":"10.1016/j.ejc.2025.104330","url":null,"abstract":"<div><div>The notion of first order convergence of graphs unifies the notions of convergence for sparse and dense graphs. Nešetřil and Ossona de Mendez (2019) proved that every first order convergent sequence of graphs from a nowhere-dense class of graphs has a modeling limit and conjectured the existence of such modeling limits with an additional property, the strong finitary mass transport principle. The existence of modeling limits satisfying the strong finitary mass transport principle was proved for first order convergent sequences of trees by Nešetřil and Ossona de Mendez (2016) and for first order sequences of graphs with bounded path-width by Gajarský et al. (2017). We establish the existence of modeling limits satisfying the strong finitary mass transport principle for first order convergent sequences of graphs with bounded tree-width.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"134 ","pages":"Article 104330"},"PeriodicalIF":0.9,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146038331","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 : 2026-04-01Epub Date: 2026-02-04DOI: 10.1016/j.ejc.2026.104348
Houshan Fu
<div><div>Jaeger et al. in 1992 introduced group coloring as the dual concept to group connectivity in graphs. Let <span><math><mi>A</mi></math></span> be an additive Abelian group, <span><math><mrow><mi>f</mi><mo>:</mo><mi>E</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>→</mo><mi>A</mi></mrow></math></span> and <span><math><mi>D</mi></math></span> an orientation of a graph <span><math><mi>G</mi></math></span>. A vertex coloring <span><math><mrow><mi>c</mi><mo>:</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>→</mo><mi>A</mi></mrow></math></span> is an <span><math><mrow><mo>(</mo><mi>A</mi><mo>,</mo><mi>f</mi><mo>)</mo></mrow></math></span>-coloring if <span><math><mrow><mi>c</mi><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow><mo>−</mo><mi>c</mi><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mo>≠</mo><mi>f</mi><mrow><mo>(</mo><mi>e</mi><mo>)</mo></mrow></mrow></math></span> for each oriented edge <span><math><mrow><mi>e</mi><mo>=</mo><mi>u</mi><mi>v</mi></mrow></math></span> from <span><math><mi>u</mi></math></span> to <span><math><mi>v</mi></math></span> under <span><math><mi>D</mi></math></span>. Kochol recently introduced the assigning polynomial to count nowhere-zero chains in graphs-nonhomogeneous analogues of nowhere-zero flows in Kochol (2022), and later extended the approach to regular matroids in Kochol (2024). Motivated by Kochol’s work, we define the <span><math><mi>α</mi></math></span>-compatible graph and the cycle-assigning polynomial <span><math><mrow><mi>P</mi><mrow><mo>(</mo><mi>G</mi><mo>,</mo><mi>α</mi><mo>;</mo><mi>k</mi><mo>)</mo></mrow></mrow></math></span> at <span><math><mi>k</mi></math></span> in terms of <span><math><mi>α</mi></math></span>-compatible spanning subgraphs, where <span><math><mi>α</mi></math></span> is an assigning of <span><math><mi>G</mi></math></span> from its cycles to <span><math><mrow><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>}</mo></mrow></math></span>. We prove that <span><math><mrow><mi>P</mi><mrow><mo>(</mo><mi>G</mi><mo>,</mo><mi>α</mi><mo>;</mo><mi>k</mi><mo>)</mo></mrow></mrow></math></span> evaluates the number of <span><math><mrow><mo>(</mo><mi>A</mi><mo>,</mo><mi>f</mi><mo>)</mo></mrow></math></span>-colorings of <span><math><mi>G</mi></math></span> for any Abelian group <span><math><mi>A</mi></math></span> of order <span><math><mi>k</mi></math></span> and <span><math><mrow><mi>f</mi><mo>:</mo><mi>E</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>→</mo><mi>A</mi></mrow></math></span> such that the assigning <span><math><msub><mrow><mi>α</mi></mrow><mrow><mi>D</mi><mo>,</mo><mi>f</mi></mrow></msub></math></span> given by <span><math><mi>f</mi></math></span> equals <span><math><mi>α</mi></math></span>. Such an assigning is admissible. Based on Kochol’s work, we derive that <span><math><mrow><msup><mrow><mi>k</mi></mrow><mrow><mo>−</mo><mi>c</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></msup><mi>P</mi><mrow><mo>(</mo><mi>G</mi><mo>,</mo><mi>α</mi><mo>;</mo><mi>k</mi><m
Jaeger et al.（1992）将群着色作为图中群连通性的对偶概念引入。设A是一个加性阿别群，f:E(G)→A， D是图G的一个取向。在D下，对于从u到V的每个有向边E =uv，如果c(V)−c(u)≠f(E)，顶点着色c:V(G)→A是一个（A,f）着色。Kochol最近在Kochol（2022）中引入了计算图中无零链的赋值多项式——无零流的非齐次类似物（Kochol），后来在Kochol（2024）中将该方法推广到正则拟阵。根据Kochol的工作，我们定义了α-相容图和循环分配多项式P（G,α;k）在k处的α-相容生成子图，其中α是G从其循环到{0,1}的分配。证明了P（G,α;k）对任意k阶阿贝尔群A和f:E(G)→A求G的(A,f)-着色的个数，使得赋值由f给出的α d，f等于α。这样的分配是可以接受的。基于Kochol的工作，我们推导出k−c(G)P（G,α;k）是一个枚举(a,f)-张力和计数特定无零链的多项式。进一步，将Whitney的破环概念推广到破相容环，证明了与可容许赋值α相关的P（G,α;k）中k|V(G)|−i的系数的绝对值等于有i条边且不包含破相容环的α-相容生成子图的个数。根据组合解释，建立了从可容许赋值到循环赋值多项式的统一保序关系，并进一步证明了对于每个环e，当α(e)=1时，对于G的任意可容许赋值α， P（G,α;k）的系数非零且符号交替。
{"title":"Polynomials counting group colorings in graphs","authors":"Houshan Fu","doi":"10.1016/j.ejc.2026.104348","DOIUrl":"10.1016/j.ejc.2026.104348","url":null,"abstract":"<div><div>Jaeger et al. in 1992 introduced group coloring as the dual concept to group connectivity in graphs. Let <span><math><mi>A</mi></math></span> be an additive Abelian group, <span><math><mrow><mi>f</mi><mo>:</mo><mi>E</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>→</mo><mi>A</mi></mrow></math></span> and <span><math><mi>D</mi></math></span> an orientation of a graph <span><math><mi>G</mi></math></span>. A vertex coloring <span><math><mrow><mi>c</mi><mo>:</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>→</mo><mi>A</mi></mrow></math></span> is an <span><math><mrow><mo>(</mo><mi>A</mi><mo>,</mo><mi>f</mi><mo>)</mo></mrow></math></span>-coloring if <span><math><mrow><mi>c</mi><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow><mo>−</mo><mi>c</mi><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mo>≠</mo><mi>f</mi><mrow><mo>(</mo><mi>e</mi><mo>)</mo></mrow></mrow></math></span> for each oriented edge <span><math><mrow><mi>e</mi><mo>=</mo><mi>u</mi><mi>v</mi></mrow></math></span> from <span><math><mi>u</mi></math></span> to <span><math><mi>v</mi></math></span> under <span><math><mi>D</mi></math></span>. Kochol recently introduced the assigning polynomial to count nowhere-zero chains in graphs-nonhomogeneous analogues of nowhere-zero flows in Kochol (2022), and later extended the approach to regular matroids in Kochol (2024). Motivated by Kochol’s work, we define the <span><math><mi>α</mi></math></span>-compatible graph and the cycle-assigning polynomial <span><math><mrow><mi>P</mi><mrow><mo>(</mo><mi>G</mi><mo>,</mo><mi>α</mi><mo>;</mo><mi>k</mi><mo>)</mo></mrow></mrow></math></span> at <span><math><mi>k</mi></math></span> in terms of <span><math><mi>α</mi></math></span>-compatible spanning subgraphs, where <span><math><mi>α</mi></math></span> is an assigning of <span><math><mi>G</mi></math></span> from its cycles to <span><math><mrow><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>}</mo></mrow></math></span>. We prove that <span><math><mrow><mi>P</mi><mrow><mo>(</mo><mi>G</mi><mo>,</mo><mi>α</mi><mo>;</mo><mi>k</mi><mo>)</mo></mrow></mrow></math></span> evaluates the number of <span><math><mrow><mo>(</mo><mi>A</mi><mo>,</mo><mi>f</mi><mo>)</mo></mrow></math></span>-colorings of <span><math><mi>G</mi></math></span> for any Abelian group <span><math><mi>A</mi></math></span> of order <span><math><mi>k</mi></math></span> and <span><math><mrow><mi>f</mi><mo>:</mo><mi>E</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>→</mo><mi>A</mi></mrow></math></span> such that the assigning <span><math><msub><mrow><mi>α</mi></mrow><mrow><mi>D</mi><mo>,</mo><mi>f</mi></mrow></msub></math></span> given by <span><math><mi>f</mi></math></span> equals <span><math><mi>α</mi></math></span>. Such an assigning is admissible. Based on Kochol’s work, we derive that <span><math><mrow><msup><mrow><mi>k</mi></mrow><mrow><mo>−</mo><mi>c</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></msup><mi>P</mi><mrow><mo>(</mo><mi>G</mi><mo>,</mo><mi>α</mi><mo>;</mo><mi>k</mi><m","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"134 ","pages":"Article 104348"},"PeriodicalIF":0.9,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146189010","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 : 2026-04-01Epub Date: 2026-02-03DOI: 10.1016/j.ejc.2026.104346
Viresh Patel , Mehmet Akif Yıldız
We consider the problem of decomposing the edges of a digraph into as few paths as possible. A natural lower bound for the number of paths in any path decomposition of a digraph is ; any digraph that achieves this bound is called consistent. Alspach et al. 1976 conjectured in 1976 that every tournament of even order is consistent and this was recently verified for large tournaments by Girão et al. 2023. A more general conjecture of Pullman (Reid and Wayland, 1987) states that for odd , every orientation of a -regular graph is consistent. We prove that the conjecture holds for random -regular graphs with high probability i.e. for fixed odd and as the conjecture holds for almost all -regular graphs. Along the way, we verify Pullman’s conjecture for graphs whose girth is sufficiently large (as a function of the degree).
我们考虑将有向图的边分解成尽可能少的路径的问题。有向图D的任意路径分解中路径数的自然下界为12∑v∈v (D)| D +(v)−D−(v)|；任何达到这个边界的有向图都被称为一致的。Alspach et al. 1976在1976年推测每个偶数顺序的锦标赛都是一致的，最近由gir等人在2023年验证了这一点。Pullman （Reid and Wayland, 1987）的一个更一般的猜想指出，对于奇数d， d正则图的每个方向都是一致的。我们证明了该猜想对高概率随机d-正则图成立，即对于固定奇数d，当n→∞时，该猜想对几乎所有d-正则图成立。在此过程中，我们验证了对于周长足够大的图（作为度的函数）的Pullman猜想。
{"title":"Path decompositions of oriented graphs","authors":"Viresh Patel , Mehmet Akif Yıldız","doi":"10.1016/j.ejc.2026.104346","DOIUrl":"10.1016/j.ejc.2026.104346","url":null,"abstract":"<div><div>We consider the problem of decomposing the edges of a digraph into as few paths as possible. A natural lower bound for the number of paths in any path decomposition of a digraph <span><math><mi>D</mi></math></span> is <span><math><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><msub><mrow><mo>∑</mo></mrow><mrow><mi>v</mi><mo>∈</mo><mi>V</mi><mrow><mo>(</mo><mi>D</mi><mo>)</mo></mrow></mrow></msub><mrow><mo>|</mo><msup><mrow><mi>d</mi></mrow><mrow><mo>+</mo></mrow></msup><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow><mo>−</mo><msup><mrow><mi>d</mi></mrow><mrow><mo>−</mo></mrow></msup><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow><mo>|</mo></mrow></mrow></math></span>; any digraph that achieves this bound is called consistent. Alspach et al. 1976 conjectured in 1976 that every tournament of even order is consistent and this was recently verified for large tournaments by Girão et al. 2023. A more general conjecture of Pullman (Reid and Wayland, 1987) states that for odd <span><math><mi>d</mi></math></span>, every orientation of a <span><math><mi>d</mi></math></span>-regular graph is consistent. We prove that the conjecture holds for random <span><math><mi>d</mi></math></span>-regular graphs with high probability i.e. for fixed odd <span><math><mi>d</mi></math></span> and as <span><math><mrow><mi>n</mi><mo>→</mo><mi>∞</mi></mrow></math></span> the conjecture holds for almost all <span><math><mi>d</mi></math></span>-regular graphs. Along the way, we verify Pullman’s conjecture for graphs whose girth is sufficiently large (as a function of the degree).</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"134 ","pages":"Article 104346"},"PeriodicalIF":0.9,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146189013","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 : 2026-04-01Epub Date: 2026-02-10DOI: 10.1016/j.ejc.2026.104350
Thomas Karam
We prove that for every integer there exists a dense collection of subsets of such that no two of them have a symmetric difference that may be written as the th power of a union of at most intervals. This provides a limitation on reasonable tightenings of a question of Alon from 2023 and of a conjecture of Gowers from 2009, and investigates a direction analogous to that of recent works of Conlon, Kamčev, Leader, Räty and Spiegel on intervals in the Hales–Jewett theorem.
{"title":"Unions of intervals in codes based on powers of sets","authors":"Thomas Karam","doi":"10.1016/j.ejc.2026.104350","DOIUrl":"10.1016/j.ejc.2026.104350","url":null,"abstract":"<div><div>We prove that for every integer <span><math><mrow><mi>d</mi><mo>≥</mo><mn>2</mn></mrow></math></span> there exists a dense collection of subsets of <span><math><msup><mrow><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow></mrow><mrow><mi>d</mi></mrow></msup></math></span> such that no two of them have a symmetric difference that may be written as the <span><math><mi>d</mi></math></span>th power of a union of at most <span><math><mrow><mo>⌊</mo><mi>d</mi><mo>/</mo><mn>2</mn><mo>⌋</mo></mrow></math></span> intervals. This provides a limitation on reasonable tightenings of a question of Alon from 2023 and of a conjecture of Gowers from 2009, and investigates a direction analogous to that of recent works of Conlon, Kamčev, Leader, Räty and Spiegel on intervals in the Hales–Jewett theorem.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"134 ","pages":"Article 104350"},"PeriodicalIF":0.9,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146189008","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 : 2026-03-01Epub 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":"2026-03-01","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 : 2026-03-01Epub 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":"2026-03-01","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 : 2026-03-01Epub Date: 2025-12-26DOI: 10.1016/j.ejc.2025.104323
Raphael Yuster
For a digraph , let be the maximum chromatic number of an acyclic subgraph of . For an -vertex digraph it is proved that where is the bipartite independence number of , i.e., the largest for which there are two disjoint -sets of vertices with no edge between them. This generalizes a result of Fox, Kwan and Sudakov, who proved this for the case (i.e., tournaments and semicomplete digraphs). Consequently, if , then which polynomially improves the folklore bound . As a corollary, with high probability, all orientations of the random -vertex graph with edge probability (in particular, constant , hence almost all -vertex graphs) satisfy . Our proof uses a theorem of Gallai and Milgram that together with several additional ideas, essentially reduces to the proof of Fox, Kwan and Sudakov.
{"title":"Acyclic subgraphs of digraphs with high chromatic number","authors":"Raphael Yuster","doi":"10.1016/j.ejc.2025.104323","DOIUrl":"10.1016/j.ejc.2025.104323","url":null,"abstract":"<div><div>For a digraph <span><math><mi>G</mi></math></span>, let <span><math><mrow><mi>f</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> be the maximum chromatic number of an acyclic subgraph of <span><math><mi>G</mi></math></span>. For an <span><math><mi>n</mi></math></span>-vertex digraph <span><math><mi>G</mi></math></span> it is proved that <span><math><mrow><mi>f</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≥</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>5</mn><mo>/</mo><mn>9</mn><mo>−</mo><mi>o</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></msup><msup><mrow><mi>s</mi></mrow><mrow><mo>−</mo><mn>14</mn><mo>/</mo><mn>9</mn></mrow></msup></mrow></math></span> where <span><math><mi>s</mi></math></span> is the bipartite independence number of <span><math><mi>G</mi></math></span>, i.e., the largest <span><math><mi>s</mi></math></span> for which there are two disjoint <span><math><mi>s</mi></math></span>-sets of vertices with no edge between them. This generalizes a result of Fox, Kwan and Sudakov, who proved this for the case <span><math><mrow><mi>s</mi><mo>=</mo><mn>0</mn></mrow></math></span> (i.e., tournaments and semicomplete digraphs). Consequently, if <span><math><mrow><mi>s</mi><mo>=</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>o</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></msup></mrow></math></span>, then <span><math><mrow><mi>f</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≥</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>5</mn><mo>/</mo><mn>9</mn><mo>−</mo><mi>o</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></msup></mrow></math></span> which polynomially improves the folklore bound <span><math><mrow><mi>f</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≥</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn><mo>−</mo><mi>o</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></msup></mrow></math></span>. As a corollary, with high probability, all orientations of the random <span><math><mi>n</mi></math></span>-vertex graph with edge probability <span><math><mrow><mi>p</mi><mo>=</mo><msup><mrow><mi>n</mi></mrow><mrow><mo>−</mo><mi>o</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></msup></mrow></math></span> (in particular, constant <span><math><mi>p</mi></math></span>, hence almost all <span><math><mi>n</mi></math></span>-vertex graphs) satisfy <span><math><mrow><mi>f</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≥</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>5</mn><mo>/</mo><mn>9</mn><mo>−</mo><mi>o</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></msup></mrow></math></span>. Our proof uses a theorem of Gallai and Milgram that together with several additional ideas, essentially reduces to the proof of Fox, Kwan and Sudakov.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"133 ","pages":"Article 104323"},"PeriodicalIF":0.9,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145840443","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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