European Journal of Combinatorics最新文献

英文中文

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 : 2026-02-01 Epub 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

A proof of some conjectures of Garvan on partitions rank and crank inequalities 关于分区秩不等式和曲柄不等式的Garvan猜想的证明

IF 0.9 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2026-02-01 Epub Date: 2025-10-09 DOI: 10.1016/j.ejc.2025.104253

Renrong Mao, Jie Huang, Fan Yang

In 1988, Garvan made conjectures on inequalities satisfied by ranks and cranks modulo 5 and 7. We obtain improvements to two of these inequalities in this paper.

1988年，Garvan对以5和7为模的秩和曲柄所满足的不等式作了猜想。本文对其中两个不等式作了改进。

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 : 2026-02-01 Epub 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 : 2026-02-01 Epub 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 : 2026-02-01 Epub 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

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 : 2026-02-01 Epub 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 : 2026-02-01 Epub 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

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 : 2026-02-01 Epub 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

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 : 2026-02-01 Epub 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

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 : 2026-02-01 Epub 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":"2026-02-01","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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