European Journal of Combinatorics最新文献

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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。

引用次数: 0

A bijection for descent sets of permutations with only even and only odd cycles 只有偶圈和奇圈的置换下降集的双射

IF 0.9 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2025-11-10 DOI: 10.1016/j.ejc.2025.104280

Sergi Elizalde

It is known that, when

n

is even, the number of permutations of

{1, 2, \dots, n}

all of whose cycles have odd length equals the number of those all of whose cycles have even length. Adin, Hegedűs and Roichman recently found a surprising refinement of this identity. They showed that, for any fixed set

J

, the equality still holds when restricting to permutations with descent set

J

on one side, and permutations with ascent set

J

on the other. Their proof uses generating functions for higher Lie characters, and it also yields a version for odd

n

. Here we give a bijective proof of their result. We first use known bijections, due to Gessel, Reutenauer and others, to restate the identity in terms of multisets of necklaces, which we interpret as words, and then describe a new weight-preserving bijection between words all of whose Lyndon factors have odd length and are distinct, and words all of whose Lyndon factors have even length. We also show that the corresponding equality about Lyndon factorizations has a short proof using generating functions.

已知，当n为偶数时，{1,2，…，n}所有循环长度为奇数的排列数等于所有循环长度为偶数的排列数。Adin， Hegedűs和Roichman最近发现了这个身份的一个令人惊讶的改进。他们证明了，对于任意固定集合J，当约束为一边是下降集合J的排列，另一边是上升集合J的排列时，等式仍然成立。他们的证明使用了更高Lie字符的生成函数，并且它也产生了奇数n的版本。这里我们给出了他们结果的一个客观证明。我们首先使用已知的双射，由于Gessel， Reutenauer和其他人，以多组项链的形式重述身份，我们将其解释为单词，然后在所有林登因子长度为奇数且不同的单词和所有林登因子长度为偶数的单词之间描述一个新的保权双射。我们还利用生成函数证明了Lyndon分解的相应等式。

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引用次数: 0

On integral variations for roots of the Laplacian matching polynomial of graphs 图的拉普拉斯匹配多项式的根的积分变分

IF 0.9 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2025-11-10 DOI: 10.1016/j.ejc.2025.104284

Yi Wang , Hai-Jian Cui , Sebastian M. Cioabă

In this paper, we study the Laplacian matching polynomial of a graph and the effect of adding edges to a graph on the roots (called Laplacian matching roots) of this polynomial. In particular, we investigate the conditions under which the Laplacian matching roots change by integer values. We prove that the Laplacian matching root integral variation in one place is impossible and the Laplacian matching root integral variation in two places is also impossible under some constraints.

本文研究了图的拉普拉斯匹配多项式，以及在图上添加边对该多项式的根（称为拉普拉斯匹配根）的影响。特别地，我们研究了拉普拉斯匹配根随整数值变化的条件。在一定的约束条件下，证明了拉普拉斯匹配根积分在一处不可能发生变化，拉普拉斯匹配根积分在两处也不可能发生变化。

引用次数: 0

Generalized Ramsey numbers of cycles, paths, and hypergraphs 环、路径和超图的广义Ramsey数

IF 0.9 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2025-11-09 DOI: 10.1016/j.ejc.2025.104281

Deepak Bal , Patrick Bennett , Emily Heath , Shira Zerbib

Given a

k

-uniform hypergraph

G

and a set of

k

-uniform hypergraphs

H

, the generalized Ramsey number

f (G, H, q)

is the minimum number of colors needed to edge-color

G

so that every copy of every hypergraph

H \in H

G

receives at least

q

different colors. In this note we obtain bounds, some asymptotically sharp, on several generalized Ramsey numbers, when

G = K_{n}

G = K_{n, n}

and

H

is a set of cycles or paths, and when

G = K_{n}^{k}

and

H

contains a clique on

k + 2

vertices or a tight cycle.

给定一个k-均匀超图G和一组k-均匀超图H，广义拉姆齐数f（G，H,q）是使G中的每个超图H∈H的每个副本至少接收到q种不同颜色所需的最小颜色数。当G=Kn或G=Kn，n和H是一组环或路径，当G=Knk和H在k+2个顶点或紧环上包含团时，我们得到了几个广义Ramsey数上的一些渐近尖锐的界。

引用次数: 0

Quasi-fixed points of substitutive systems 替代系统的拟不动点

IF 0.9 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2025-11-08 DOI: 10.1016/j.ejc.2025.104282

Elżbieta Krawczyk

We study automatic sequences and automatic systems generated by general constant length (nonprimitive) substitutions. While an automatic system is typically uncountable, the set of automatic sequences is countable, implying that most sequences within an automatic system are not themselves automatic. We provide a complete and succinct classification of automatic sequences that lie in a given automatic system in terms of the quasi-fixed points of the substitution defining the system. Our result extends to factor maps between automatic systems and highlights arithmetic properties underpinning these systems. We conjecture that a similar statement holds for general nonconstant length substitutions.

研究了由一般定长（非原语）替换生成的自动序列和自动系统。虽然自动系统通常是不可数的，但自动序列的集合是可数的，这意味着自动系统中的大多数序列本身不是自动的。根据定义系统的代换的拟不动点，给出了给定自动系统中自动序列的一个完整而简洁的分类。我们的结果扩展到自动系统之间的因子映射，并突出了支撑这些系统的算术特性。我们推测，对于一般的非恒定长度替换，也有类似的结论。

引用次数: 0

On the structures of subset sums in higher dimension 高维子集和的结构

IF 0.9 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2025-11-04 DOI: 10.1016/j.ejc.2025.104279

Norbert Hegyvári , Máté Pálfy , Erfei Yue

A given subset

A

of natural numbers is said to be complete if every sufficiently large integer of

N

is the sum of distinct terms taken from

A

. In higher dimension the definition is similar: for any

X = {x_{1}, x_{2}, \dots} \subseteq N^{k}

let

F S (X) ≔ {\sum_{i = 1}^{\infty} ɛ_{i} x_{i} : x_{i} \in X, ɛ_{i} \in {0, 1}, \sum_{i = 1}^{\infty} ɛ_{i} < \infty} .

We say that a set

X

is complete respect to the region

R \subseteq N^{k}

R \subseteq F S (X)

holds. A set

X

is a thin complete set of

R

if the counting function

X (N) \leq k {log}_{2} R (N) + t_{X}

for some

t_{X}

and

F S (X) \supseteq R

. We construct ‘thin’ complete set provided the domain

R

does not contain half-lines parallel to the axis. Furthermore we investigate the distribution of the subset sum of ‘splitable’ sets too.

如果N的每一个足够大的整数是取自A的不同项的和，则给定自然数子集A是完备的：在高维中，其定义类似：对于任意X={x1,x2，…},≥Nk，令FS(X)是{∑i=1∞,i∈X, i∈{0,1},∑i=1∞,i<∞}。我们说，如果R∈FS(X)成立，则集合X对于区域R⊥k是完全的。如果某个tX和FS(X)的计数函数X(N)≤klog2R(N)+tX，则集合X是R的瘦完全集。如果定义域R不包含平行于轴的半直线，则构造“薄”完备集。进一步研究了可分集的子集和的分布。

引用次数: 0

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