European Journal of Combinatorics最新文献

英文中文

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。

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引用次数: 0

Erdős–Ko–Rado type results for partitions via spread approximations Erdős-Ko-Rado通过扩展近似为分区键入结果

IF 0.9 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2025-11-21 DOI: 10.1016/j.ejc.2025.104288

Andrey Kupavskii

In this paper, we address several Erdős–Ko–Rado type questions for families of partitions. Two partitions of

[n]

are

t

-intersecting if they share at least

t

parts, and are partially

t

-intersecting if some of their parts intersect in at least

t

elements. The question of what is the largest family of pairwise

t

-intersecting partitions was studied for several classes of partitions: Peter Erdős and Székely studied partitions of

[n]

into

ℓ

parts of unrestricted size; Ku and Renshaw studied unrestricted partitions of

[n]

; Meagher and Moura, and then Godsil and Meagher studied partitions into

ℓ

parts of equal size. We improve and generalize the results proved by these authors.

Meagher and Moura, following the work of Erdős and Székely, introduced the notion of partially

t

-intersecting partitions, and conjectured, what should be the largest partially

t

-intersecting family of partitions into

ℓ

parts of equal size

k

. The main result of this paper is the proof of their conjecture for all

t, k

, provided

ℓ

is sufficiently large.

All our results are applications of the spread approximation technique, introduced by Zakharov and the author. In order to use it, we need to refine some of the theorems from the original paper. As a byproduct, this makes the present paper a self-contained presentation of the spread approximation technique for

t

-intersecting problems.

在本文中，我们为分区族解决了几个Erdős-Ko-Rado类型问题。如果[n]的两个分区至少有t个部分是t相交的，如果它们的某些部分在至少t个元素中相交，则是部分t相交的。对于几类分区，我们研究了最大的两两t相交分区族是什么：Peter Erdős和sz kely研究了[n]划分为无限制大小的r部分的分区；Ku和Renshaw研究了[n]的无限制分区；Meagher和Moura，然后Godsil和Meagher研究了等长的分区。我们改进和推广了这些作者所证明的结果。Meagher和Moura，继Erdős和szeminkely的工作之后，引入了部分t相交分区的概念，并推测了最大的部分t相交分区族应该是多少，这些分区分为大小相等的k个部分。本文的主要结果是证明了他们对所有t，k的猜想，假设r足够大。我们所有的结果都是应用了扎哈罗夫和作者介绍的扩散近似技术。为了使用它，我们需要改进原论文中的一些定理。作为一个副产品，这使得本文成为t-相交问题的扩展逼近技术的一个完整的表述。

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引用次数: 0

Critically intersecting hypergraphs 临界相交超图

IF 0.9 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2025-11-21 DOI: 10.1016/j.ejc.2025.104286

P. Frankl

Let

k > t \geq 1

and

d = k - t

. The complete

k

-graph on

k + d

vertices has

(\frac{k + d}{d})

edges, any two edges intersect in at least

t

-vertices. Moreover, there is no

(k - 1)

-element set intersecting each edge in at least

t

vertices. The main result shows that for

k \geq d^{4}

any

k

-graph with the above properties has at most

(\frac{k + d}{d})

edges and the complete

k

-graph is the unique optimal family.

令k>；t≥1,d=k−t。k+d个顶点上的完备k图有k+dd条边，任意两条边至少相交t个顶点。而且，不存在至少t个顶点与每条边相交的（k−1）元素集。主要结果表明，当k≥d4时，具有上述性质的k图最多有k+dd条边，且完全k图是唯一最优族。

引用次数: 0

The Critical Index of Brouwer’s conjecture 布劳威尔猜想的临界指数

IF 0.9 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2025-11-18 DOI: 10.1016/j.ejc.2025.104287

Guilherme Simon Torres , Vilmar Trevisan

For a graph

G

having Laplacian spectrum

μ_{1} \geq μ_{2} \geq \dots \geq μ_{n} = 0

, Brouwer’s Conjecture states that

S_{k} (G) = \sum_{i = 1}^{k} μ_{i} \leq m + (\frac{k + 1}{2}), for any 1 \leq k \leq n

. We prove that for each graph

G

, there is a number

h \in {1, \dots, n}

, called Brouwer Critical Index - BCI, so that if

S_{h} (G) \leq m + (\frac{h + 1}{2})

, then

G

satisfies Brouwer’s Conjecture. We also explore this graph invariant as a spectral parameter, obtaining natural properties. As an application of the BCI, we show that a class of bipartite graphs satisfies Brouwer’s Conjecture. Additionally, we prove that the corona product of graphs preserves Brouwer’s Conjecture.

对于拉普拉斯谱μ1≥μ2≥⋯≥μn=0的图G， browwer猜想指出，对于任意1≤k≤n， Sk(G)=∑i=1kμi≤m+k+12。证明了对于每一个图G，存在一个数h∈{1，…，n}，称为browwer临界指数- BCI，使得当Sh(G)≤m+h+12，则G满足browwer猜想。我们也探索了这个图不变量作为谱参数，得到了自然性质。作为BCI的一个应用，我们证明了一类二部图满足Brouwer猜想。此外，我们还证明了图的冕积保持了browwer猜想。

引用次数: 0

The acyclic directed bunkbed conjecture is false 无环有向铺层猜想是假的

IF 0.9 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2025-11-18 DOI: 10.1016/j.ejc.2025.104289

Tomasz Przybyłowski

We construct a simple acyclic directed graph for which the Bunkbed Conjecture is false, thereby resolving conjectures posed by Leander and by Hollom.

我们构造了一个简单的无环有向图，它的铺床猜想是假的，从而解决了Leander和Hollom的猜想。

引用次数: 0

An identity relating Catalan numbers to tangent numbers with arithmetic applications 加泰罗尼亚数与正切数之间的恒等式及其算术应用

IF 0.9 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2025-11-13 DOI: 10.1016/j.ejc.2025.104283

Tongyuan Zhao , Zhicong Lin , Yongchun Zang

We prove a combinatorial identity relating Catalan numbers to tangent numbers arising from the study of peak algebra that was conjectured by Aliniaeifard and Li. This identity leads to the discovery of the intriguing identity

\sum_{k = 0}^{n - 1} (\frac{2 n}{2 k + 1}) 2^{2 n - 2 k} {(- 1)}^{k} E_{2 k + 1} = 2^{2 n + 1},

where

E_{2 k + 1}

denote the tangent numbers. Interestingly, the latter identity can be applied to prove that

(n + 1) E_{2 n + 1}

is divisible by

2^{2 n}

and the quotient is an odd number, a fact whose traditional proofs require significant calculations. Moreover, we find a natural

q

-analog of the latter identity with a combinatorial proof. This

q

-identity can be applied to prove Foata’s divisibility property of the

q

-tangent numbers, which responds to a problem raised by Schützenberger.

我们证明了由Aliniaeifard和Li在峰代数研究中提出的Catalan数与正切数的组合恒等式。这个恒等式引出了一个有趣的恒等式∑k=0n−12n2k+122n−2k(−1)kE2k+1=22n+1，其中E2k+1表示正切数。有趣的是，后一个恒等式可以用来证明（n+1）E2n+1可以被22n整除，并且商是奇数，这一事实的传统证明需要大量的计算。此外，我们用组合证明找到了后一个恒等式的自然q-类似。这个q-恒等式可以用来证明q-切数的Foata可除性，从而回答了sch岑伯格提出的一个问题。

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