European Journal of Combinatorics最新文献

英文中文

Bounded degree graphs and hypergraphs with no full rainbow matchings 没有完全彩虹匹配的有界度图和超图

IF 0.9 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2025-12-16 DOI: 10.1016/j.ejc.2025.104316

Ronen Wdowinski

Given a multi-hypergraph

G

that is edge-colored into color classes

E_{1}, \dots, E_{n}

, a full rainbow matching is a matching of

G

that contains exactly one edge from each color class

E_{i}

. One way to guarantee the existence of a full rainbow matching is to have the size of each color class

E_{i}

be sufficiently large compared to the maximum degree of

G

. 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

r \geq 1

and

Δ \geq 2

, we construct edge-colored

r

-uniform multi-hypergraphs with maximum degree

Δ

such that each color class has size

| E_{i} | \geq r Δ - 1

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。

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引用次数: 0

Hypercube minor-universality 超立方体minor-universality

IF 0.9 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2025-12-16 DOI: 10.1016/j.ejc.2025.104317

Itai Benjamini, Or Bernard Kalifa, Elad Tzalik

A graph

G

m

-minor-universal if every graph with at most

m

edges (and no isolated vertices) is a minor of

G

. We prove that the

d

-dimensional hypercube,

Q_{d}

, is

Ω (\frac{2^{d}}{d})

-minor-universal, and that there exists an absolute constant

C > 0

such that

Q_{d}

is not

\frac{C 2^{d}}{\sqrt{d}}

-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

n_{1}, \dots, n_{d}

be positive integers, and define

X ≔ [n_{1}] \times \dots \times [n_{d}]

. We prove that every permutation

σ : X \to X

can be expressed as

σ = σ_{1} \circ \dots \circ σ_{2 d - 1}

, where each

σ_{i}

is a one-dimensional permutation, meaning it fixes all coordinates except possibly one. We discuss future directions and pose open problems.

如果图G最多有m条边（没有孤立顶点）的图是G的次元，则图G是m次泛泛的。我们证明了d维超立方体Qd是Ω2dd-minor-universal，并且存在一个绝对常数C>；0使得Qd不是c2dd -次泛泛的。在一个更一般的情况下也得到了类似的结果，在这里我们限定了有限连通图的乘积中的子向量的大小。我们的证明的一个关键部分是关于将一个盒子的排列分解为更简单的一维排列的声明：设n1，…，和为正整数，并定义X是[n1]×⋯×[nd]。我们证明了每个排列σ:X→X都可以表示为σ=σ1°σ2d - 1，其中每个σi是一个一维排列，这意味着它固定了除了可能的一个坐标之外的所有坐标。我们讨论未来的方向，并提出悬而未决的问题。

引用次数: 0

Degree-truncated DP-colourability of K2,4-minor-free graphs K2,4次元无图的度截断dp -可色性

IF 0.9 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2025-12-12 DOI: 10.1016/j.ejc.2025.104306

On-Hei Solomon Lo , Cheng Wang , Huan Zhou , Xuding Zhu

Assume

G

is a graph and

k

is a positive integer. Let

f : V (G) \to N

be defined as

f (v) = min {k, d_{G} (v)}

. If

G

is DP-

f

-colourable (respectively,

f

-choosable), then we say

G

is degree-truncated DP-

k

-colourable (respectively, degree-truncated

k

-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

K_{2, 4}

-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.

假设G是一个图，k是一个正整数。设f:V(G)→N定义为f(V)=min{k,dG(V)}。如果G是dp -f可着色的（分别为f-可选择的），那么我们说G是度截断的dp -k可着色的（分别为度截断的k-可选择的）。Hutchinson（2008）证明了除三角形以外的2连通极大外平面图是度截断5-可选的。Hutchinson问这个结果是否可以推广到所有的外平面图。证明了除环图和完全图以外的2连通K2图、4次自由图是度截断的dp -5可色图。这不仅肯定地回答了Hutchinson的问题，而且还扩展到更大的图族，并加强了对dp -可色性的选择性。

引用次数: 0

Order polytopes of crown posets 冠位集的序多面体

IF 0.9 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2025-12-08 DOI: 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

f

-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

h^{*}

-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).

近十年来，人们对之字形背集的序多面体进行了深入的研究。一个相关的偏序集，称为冠偏序集，是通过在一个偶数之字形偏序集的端点之间添加一个额外的覆盖关系而得到的，它不是很好理解。本文研究了冠序集的序多面体。我们给出了它们的f向量的显式公式。我们提供了Ehrhart多项式的递归公式，与Petersen和Zhuang（2025）在之字形情况下发现的公式相对应。我们使用这些公式来简化Ferroni， Morales和Panova（2025）对这些偏序集的阶多项式的线性项的计算。此外，我们根据循环交替排列上的循环交换统计量提供了h * -多项式系数的组合解释，它提供了Coons和sullivan（2023）的结果的循环版本。

引用次数: 0

Unbounded matroids 无限的拟阵

IF 0.9 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2025-12-04 DOI: 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.

一个矩阵基多面体是一个多面体，其中每个顶点有0,1个坐标，每个边平行于两个坐标向量的差。用满足单位递增性质的布尔格上的积分次模函数组合描述了矩阵基多边形。通过用任意分配格代替布尔格，我们定义了一类更一般的无界拟阵，即u -拟阵。因此，u -拟阵可以作为满足拟阵基多面体顶点和边缘条件的多面体的组合模型，但可能是无界的。和多拟阵一样，u -拟阵是对拟阵的推广，是子模系统的一种特殊情况。我们证明了每一个u -矩阵都有一个正则最大扩展，我们称之为广义扩展；类似的几何表述是：每一个u型矩阵基多面体都包含一个唯一的最大的矩阵基多面体。我们证明了一个u -矩阵基多面体的顶点支撑点跨出一个可壳化的简单复体，并从壳化阶的角度对u -矩阵基系统进行了刻画，推广了一个简单复体是一个与矩阵无关的复体的Björner准则和Gale准则。最后，我们给出了我们的理论在子空间排列中的一个应用，并证明了在这种情况下，广义扩展具有自然的几何解释。

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引用次数: 0

Fragile minor-monotone parameters under a random edge perturbation 随机边缘扰动下的脆弱次单调参数

IF 0.9 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2025-12-04 DOI: 10.1016/j.ejc.2025.104305

Dong Yeap Kang , Mihyun Kang , Jaehoon Kim , Sang-il Oum

We conduct a quantitative analysis of how many random edges need to be added to a base graph

H

in order to significantly increase natural minor-monotone graph parameters of the resulting graph

R

. Specifically, we show that if

R

is obtained from a connected graph

H

by adding only a few random edges, the tree-width, genus, and Hadwiger number of

R

become very large, irrespective of the structure of

H

我们定量分析了一个基本图H中需要添加多少条随机边才能显著增加最终图R的自然次单调图参数。具体来说，我们表明，如果从连通图H中仅添加少量随机边即可获得R，则R的树宽、属数和哈德维格数将变得非常大，而与H的结构无关。

引用次数: 0

On the separating Noether number of finite abelian groups 有限阿贝尔群的分离Noether数

IF 0.9 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2025-12-03 DOI: 10.1016/j.ejc.2025.104302

Barna Schefler , Kevin Zhao , Qinghai Zhong

The separating Noether number

β_{sep} (G)

of a finite group

G

is the minimal positive integer

d

such that for every finite dimensional

G

-module

V

there is a separating set consisting of invariant polynomials of degree at most

d

. 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

β_{sep} (G)

, provided that

G

is either a

p

-group or has rank 2, 3 or 5.

有限群G的分离诺特数βsep(G)是最小正整数d，使得对于每一个有限维G模V，存在一个不超过d次的不变多项式组成的分离集。本文利用加性组合学的方法研究了有限阿贝尔群的分离诺特数。其中，我们得到了βsep(G)的精确值，假设G是p群或秩为2、3或5。

引用次数: 0

A polynomial Ramsey statement for bounded VC-dimension 有界vc维的多项式Ramsey命题

IF 0.9 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2025-12-02 DOI: 10.1016/j.ejc.2025.104303

Tomáš Hons

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.

Ding, Oporowski， Oxley和Vertigan的一个定理指出，每一个足够大的没有孪生的二部图都包含一个任意大小的匹配、共匹配或半图作为诱导子图。我们利用最近对有界vc维图的Erdős-Hajnal性质的验证，证明了假设初始图的vc维有界，这个Ramsey语句具有多项式依赖性。由于Ding等人的定理在（有限）模型理论中发挥作用，该理论研究更受限制的结构，因此我们还在此背景下对该定理的进一步改进进行了评论。

引用次数: 0

Interval hypergraphic lattices 区间超图格

IF 0.9 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2025-11-25 DOI: 10.1016/j.ejc.2025.104285

Nantel Bergeron , Vincent Pilaud

For a hypergraph

H

[n]

, the hypergraphic poset

P_{H}

is the transitive closure of the oriented skeleton of the hypergraphic polytope

△_{H}

(the Minkowski sum of the standard simplices

△_{H}

for all

H \in H

). Hypergraphic posets include the weak order for the permutahedron (when

H

is the complete graph on

[n]

) and the Tamari lattice for the associahedron (when

H

is the set of all intervals of

[n]

), which motivates the study of lattice properties of hypergraphic posets. In this paper, we focus on interval hypergraphs, where all hyperedges are intervals of

[n]

. We characterize the interval hypergraphs

I

for which

P_{I}

is a lattice, a distributive lattice, a semidistributive lattice, and a lattice quotient of the weak order.

对于[n]上的超图H，超图偏序集PH是超图多面体△H（所有H∈H的标准简形△H的Minkowski和）的有向骨架的传递闭包。超图偏序集包括了复面体的弱序（当H是[n]上的完全图时）和结合面体的Tamari格（当H是[n]上所有区间的集合时），这激发了对超图偏序集格性质的研究。在本文中，我们关注区间超图，其中所有的超边都是区间[n]。刻画了π为格、分配格、半分配格和弱阶格商的区间超图I。

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引用次数: 0

Odd covers of complete graphs and hypergraphs 完全图和超图的奇盖

IF 0.9 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2025-11-24 DOI: 10.1016/j.ejc.2025.104299

Imre Leader , Ta Sheng Tan

The ‘odd cover number’ of a complete graph is the smallest size of a family of complete bipartite graphs that covers each edge an odd number of times. For

n

odd, Buchanan, Clifton, Culver, Nie, O’Neill, Rombach and Yin showed that the odd cover number of

K_{n}

is equal to

(n + 1) / 2

(n + 3) / 2

, and they conjectured that it is always

(n + 1) / 2

. We prove this conjecture.

For

n

even, Babai and Frankl showed that the odd cover number of

K_{n}

is always at least

n / 2

, and the above authors and Radhakrishnan, Sen and Vishwanathan gave some values of

n

for which equality holds. We give some new examples.

Our constructions arise from some very symmetric constructions for the corresponding problem for complete hypergraphs. That is, the odd cover number of the complete 3-graph

K_{n}^{(3)}

is the smallest number of complete 3-partite 3-graphs such that each 3-set is in an odd number of them. We show that the odd cover number of

K_{n}^{(3)}

is exactly

n / 2

for even

n

, and we show that for odd

n

it is

(n - 1) / 2

for infinitely many values of

n

. We also show that for

r = 3

and

r = 4

, the odd cover number of

K_{n}^{(r)}

is strictly less than the partition number, answering a question of Buchanan, Clifton, Culver, Nie, O’Neill, Rombach and Yin for those values of

r

完全图的“奇覆盖数”是覆盖每条边奇数次的完全二部图族的最小尺寸。对于n个奇数，Buchanan, Clifton, Culver, Nie, O 'Neill， Rombach和Yin证明了Kn的奇数覆盖数等于(n+1)/2或(n+3)/2，并推测它总是(n+1)/2。我们证明了这个猜想。对于偶数n， Babai和Frankl证明了Kn的奇覆盖数总是至少为n/2，并且上述作者和Radhakrishnan， Sen和Vishwanathan给出了n的一些相等值。我们给出了一些新的例子。我们的构造是由完全超图对应问题的一些非常对称的构造而来的。即完全3-图Kn(3)的奇盖数是使每个3-集都在奇数个完全3-图中的最小数目。我们证明了对于偶数n， Kn(3)的奇覆盖数恰好是n/2，对于无限多个n值，我们证明了对于奇数n，它是(n−1)/2。我们还证明了对于r=3和r=4, Kn(r)的奇覆盖数严格小于分区数，回答了Buchanan， Clifton, Culver, Nie, O 'Neill， Rombach和Yin对于这些r值的问题。

{"title":"Odd covers of complete graphs and hypergraphs","authors":"Imre Leader , Ta Sheng Tan","doi":"10.1016/j.ejc.2025.104299","DOIUrl":"10.1016/j.ejc.2025.104299","url":null,"abstract":"<div><div>The ‘odd cover number’ of a complete graph is the smallest size of a family of complete bipartite graphs that covers each edge an odd number of times. For <span><math><mi>n</mi></math></span> odd, Buchanan, Clifton, Culver, Nie, O’Neill, Rombach and Yin showed that the odd cover number of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> is equal to <span><math><mrow><mrow><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mo>/</mo><mn>2</mn></mrow></math></span> or <span><math><mrow><mrow><mo>(</mo><mi>n</mi><mo>+</mo><mn>3</mn><mo>)</mo></mrow><mo>/</mo><mn>2</mn></mrow></math></span>, and they conjectured that it is always <span><math><mrow><mrow><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mo>/</mo><mn>2</mn></mrow></math></span>. We prove this conjecture.</div><div>For <span><math><mi>n</mi></math></span> even, Babai and Frankl showed that the odd cover number of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> is always at least <span><math><mrow><mi>n</mi><mo>/</mo><mn>2</mn></mrow></math></span>, and the above authors and Radhakrishnan, Sen and Vishwanathan gave some values of <span><math><mi>n</mi></math></span> for which equality holds. We give some new examples.</div><div>Our constructions arise from some very symmetric constructions for the corresponding problem for complete hypergraphs. That is, the odd cover number of the complete 3-graph <span><math><msubsup><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow><mrow><mrow><mo>(</mo><mn>3</mn><mo>)</mo></mrow></mrow></msubsup></math></span> is the smallest number of complete 3-partite 3-graphs such that each 3-set is in an odd number of them. We show that the odd cover number of <span><math><msubsup><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow><mrow><mrow><mo>(</mo><mn>3</mn><mo>)</mo></mrow></mrow></msubsup></math></span> is exactly <span><math><mrow><mi>n</mi><mo>/</mo><mn>2</mn></mrow></math></span> for even <span><math><mi>n</mi></math></span>, and we show that for odd <span><math><mi>n</mi></math></span> it is <span><math><mrow><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mo>/</mo><mn>2</mn></mrow></math></span> for infinitely many values of <span><math><mi>n</mi></math></span>. We also show that for <span><math><mrow><mi>r</mi><mo>=</mo><mn>3</mn></mrow></math></span> and <span><math><mrow><mi>r</mi><mo>=</mo><mn>4</mn></mrow></math></span>, the odd cover number of <span><math><msubsup><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow><mrow><mrow><mo>(</mo><mi>r</mi><mo>)</mo></mrow></mrow></msubsup></math></span> is strictly less than the partition number, answering a question of Buchanan, Clifton, Culver, Nie, O’Neill, Rombach and Yin for those values of <span><math><mi>r</mi></math></span>.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"132 ","pages":"Article 104299"},"PeriodicalIF":0.9,"publicationDate":"2025-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145623504","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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