European Journal of Combinatorics最新文献

英文中文

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

Twists and twistability 扭转和扭转能力

IF 0.9 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2025-10-30 DOI: 10.1016/j.ejc.2025.104259

Rebecca Coulson

Metrically homogeneous graphs are connected graphs which, when endowed with the path metric, are homogeneous as metric spaces. In this paper we introduce the concept of twisted automorphisms, a notion of isomorphism up to a permutation of the language. We find all permutations of the language which are associated with twisted automorphisms of metrically homogeneous graphs. For each non-trivial permutation of this type we also characterize the class of metrically homogeneous graphs which allow a twisted isomorphism associated with that permutation. The permutations we find are, remarkably, precisely those found by Bannai and Bannai in an analogous result in the context of finite association schemes (Bannai and Bannai, 1980), though why this might be is still an open question.

度量齐次图是连通图，当赋予路径度量时，它们作为度量空间是齐次的。在本文中，我们引入了扭曲自同构的概念，这是一种同构的概念，一直到语言的置换。我们找到了与度量齐次图的扭曲自同构相关的所有语言置换。对于这种类型的每一个非平凡置换，我们也刻画了一类允许与该置换相关联的扭曲同构的度量齐次图。值得注意的是，我们发现的排列正是Bannai和Bannai在有限关联方案背景下的类似结果中发现的排列（Bannai和Bannai， 1980），尽管为什么这可能仍然是一个悬而未决的问题。

引用次数: 0

Approximating maximum-size properly colored forests 近似最大尺寸的适当着色的森林

IF 0.9 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2025-10-27 DOI: 10.1016/j.ejc.2025.104269

Yuhang Bai , Kristóf Bérczi , Gergely Csáji , Tamás Schwarcz

In the Properly Colored Spanning Tree problem, we are given an edge-colored undirected graph and the goal is to find a properly colored spanning tree, i.e., a spanning tree in which any two adjacent edges have distinct colors. The problem is interesting not only from a graph coloring point of view, but is also closely related to the Degree Bounded Spanning Tree and

(1, 2)

-Traveling Salesman problems, two classical questions that have attracted considerable interest in combinatorial optimization and approximation theory. Previous work on properly colored spanning trees has mainly focused on determining the existence of such a tree and hence has not considered the question from an algorithmic perspective. We propose an optimization version called Maximum-size Properly Colored Forest problem, which aims to find a properly colored forest with as many edges as possible. We consider the problem in different graph classes and for different numbers of colors, and present polynomial-time approximation algorithms as well as inapproximability results for these settings. Our proof technique relies on the sum of matching matroids defined by the color classes, a connection that might be of independent combinatorial interest. We also consider the Maximum-size Properly Colored Tree problem asking for the maximum size of a properly colored tree not necessarily spanning all the vertices. We show that the optimum is significantly more difficult to approximate than in the forest case, and provide an approximation algorithm for complete multigraphs.

在适当着色生成树问题中，我们给定一个边着色的无向图，目标是找到一棵适当着色的生成树，即任意两个相邻边具有不同颜色的生成树。这个问题不仅从图着色的角度来看是有趣的，而且与度有界生成树和(1,2)-旅行商问题密切相关，这两个经典问题在组合优化和近似理论中引起了相当大的兴趣。以前关于适当着色生成树的工作主要集中在确定这种树的存在性上，因此没有从算法的角度考虑这个问题。我们提出了一个优化版本，称为最大尺寸适当着色森林问题，其目的是找到一个具有尽可能多边的适当着色森林。我们考虑了不同图类和不同颜色数量的问题，并给出了多项式时间逼近算法以及这些设置的不可逼近性结果。我们的证明技术依赖于由颜色类定义的匹配拟阵的和，这种连接可能具有独立的组合兴趣。我们还考虑了最大尺寸适当着色树问题，该问题要求适当着色树的最大尺寸，而不必跨越所有顶点。我们证明了最优的近似比在森林情况下要困难得多，并提供了一个完全多图的近似算法。

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引用次数: 0

On ω-categorical structures with few finite substructures 关于具有少数有限子结构的ω-范畴结构

IF 0.9 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2025-10-27 DOI: 10.1016/j.ejc.2025.104266

Pierre Simon

We establish new results on the possible growth rates for the sequence

(f_{n})

counting the number of orbits of a given oligomorphic group on unordered sets of size

n

. Macpherson showed that for primitive actions, the growth is at least exponential (if the sequence is not constant equal to 1). The best lower bound previously known for the base of the exponential was obtained by Merola. We establishing the optimal value of 2 in the case where the structure is unstable. This allows us to improve on Merola’s bound and also obtain the optimal value for structures homogeneous in a finite relational language. Finally, we show that the study of sequences

(f_{n})

of sub-exponential growth reduces to the

ω

-stable case.

我们建立了在大小为n的无序集合上计算给定寡纯群轨道数的序列（fn）可能增长率的新结果。Macpherson证明了对于原始动作，增长率至少是指数级的（如果序列不等于1）。以前已知的指数底的最佳下界是由梅罗拉得到的。在结构不稳定的情况下，我们建立了2的最优值。这使我们能够改进Merola界，并获得有限关系语言中齐次结构的最优值。最后，我们证明了对次指数增长序列（fn）的研究可以简化到ω稳定的情况。

引用次数: 0

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