European Journal of Combinatorics最新文献

英文中文

Hypercube minor-universality 超立方体minor-universality

IF 0.9 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2026-03-01 Epub 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

A characterization of the Grassmann graphs: One missing case 格拉斯曼图的表征：一个缺失的情况

IF 0.9 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2026-03-01 Epub Date: 2025-12-20 DOI: 10.1016/j.ejc.2025.104320

Jack H. Koolen , Chenhui Lv , Alexander L. Gavrilyuk

We prove that the Grassmann graphs

J_{2} (2 D + 3, D)

D \geq 3

, are characterized by their intersection numbers, which settles one of the few remaining cases.

我们证明了Grassmann图J2(2D+3，D)， D≥3是由它们的相交数来表征的，这解决了剩下的少数情况之一。

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 : 2026-03-01 Epub 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 : 2026-03-01 Epub 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

Generalized quasikernels in digraphs 有向图中的广义拟核

IF 0.9 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2026-03-01 Epub Date: 2025-12-20 DOI: 10.1016/j.ejc.2025.104307

Sam Spiro

Given a digraph

D

, we say that a set of vertices

Q \subseteq V (D)

is a

q

-kernel if

Q

is an independent set and if every vertex of

D

can be reached from

Q

by a path of length at most

q

. In this paper, we initiate the study of several extremal problems for

q

-kernels. For example, we introduce and make progress on (what turns out to be) a weak version of the Small Quasikernel Conjecture, namely that every digraph contains a

q

-kernel with

| N^{+} [Q] | \geq \frac{1}{2} | V (D) |

for all

q \geq 2

给定一个有向图D，如果Q是一个独立的集合，且D的每个顶点从Q出发可经一条长度不超过Q的路径到达，则称顶点集Q≥V(D)为Q核。本文研究了Q核的几个极值问题。例如，我们引入并在小拟核猜想的弱版本上取得了进展，即每个有向图都包含一个Q核，其中|N+[Q]|≥12|V(D)|对于所有Q≥2。

引用次数: 0

A spectral lower bound on chromatic numbers using p-energy 利用p能量的色数的谱下界

IF 0.9 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2026-02-01 Epub Date: 2025-10-13 DOI: 10.1016/j.ejc.2025.104252

Clive Elphick , Quanyu Tang , Shengtong Zhang

<div><div>Let <math><msub><mrow><mi>A</mi></mrow><mrow><mi>G</mi></mrow></msub></math> be the adjacency matrix of a simple graph <math><mi>G</mi></math>, and let <math><mrow><mi>χ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math>, <math><mrow><msub><mrow><mi>χ</mi></mrow><mrow><mi>f</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math>, <math><mrow><msub><mrow><mi>χ</mi></mrow><mrow><mi>q</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math>, <math><mrow><mi>ξ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math> and <math><mrow><msub><mrow><mi>ξ</mi></mrow><mrow><mi>f</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math> denote its chromatic number, fractional chromatic number, quantum chromatic number, orthogonal rank and projective rank, respectively. For <math><mrow><mi>p</mi><mo>≥</mo><mn>0</mn></mrow></math>, we define the positive and negative <math><mi>p</mi></math>-energies of <math><mi>G</mi></math> by <math><mrow><msubsup><mrow><mi>E</mi></mrow><mrow><mi>p</mi></mrow><mrow><mo>+</mo></mrow></msubsup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><munder><mrow><mo>∑</mo></mrow><mrow><msub><mrow><mi>λ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>></mo><mn>0</mn></mrow></munder><msubsup><mrow><mi>λ</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>p</mi></mrow></msubsup><mo>,</mo><mspace></mspace><msubsup><mrow><mi>E</mi></mrow><mrow><mi>p</mi></mrow><mrow><mo>−</mo></mrow></msubsup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><munder><mrow><mo>∑</mo></mrow><mrow><msub><mrow><mi>λ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo><</mo><mn>0</mn></mrow></munder><msup><mrow><mrow><mo>|</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>|</mo></mrow></mrow><mrow><mi>p</mi></mrow></msup><mo>,</mo></mrow></math>where <math><mrow><msub><mrow><mi>λ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>≥</mo><mo>⋯</mo><mo>≥</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></math> are the eigenvalues of <math><msub><mrow><mi>A</mi></mrow><mrow><mi>G</mi></mrow></msub></math>. We prove that for all <math><mrow><mi>p</mi><mo>≥</mo><mn>0</mn></mrow></math>, <math><mrow><mi>χ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≥</mo><mfenced><mrow><msub><mrow><mi>χ</mi></mrow><mrow><mi>f</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>,</mo><msub><mrow><mi>χ</mi></mrow><mrow><mi>q</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>,</mo><mi>ξ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></mfenced><mo>≥</mo><msub><mrow><mi>ξ</mi></mrow><mrow><mi>f</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≥</mo><mn>1</mn><mspace></mspace><mo>+</mo><mo>max</mo><msup

设AG为简单图G的邻接矩阵，χ(G)、χf(G)、χq(G)、ξ(G)、ξf(G)分别表示它的色数、分数色数、量子色数、正交秩和射影秩。对于p≥0，我们定义了G的正负p能：Ep+(G)=∑λi>0λip,Ep−(G)=∑λi<0|λi|p，其中λ1≥⋯≥λn为AG的特征值。我们证明所有p≥0,χ(G)≥χf (G),χq (G),ξ(G)≥ξf (G)≥1 + maxEp + (G) Ep−(G), Ep−(G) Ep + (G) 1 | p−1 |。此结果统一并强化了p∈{0,2，∞}所对应的一系列已有界。特别地，当p=0时，得到惯性界χf(G)≥ξf(G)≥1+maxn+n -,n - n+，其中n+和n -分别表示AG的正特征值和负特征值的个数。这解决了Elphick和Wocjan的两个猜想。我们还证明了对于某些图，p的非整数值提供比现有谱界更清晰的下界。作为一个例子，我们确定了Tilley图的χq，这不能使用现有的（未加权的）p-能界来实现。我们的证明采用了一种新颖的线性代数和测量理论工具的综合，这使我们能够超越现有的谱界。

引用次数: 0

Intervals in the Hales–Jewett theorem Hales-Jewett定理中的区间

IF 0.9 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2026-02-01 Epub Date: 2025-10-21 DOI: 10.1016/j.ejc.2025.104258

David Conlon , Nina Kamčev

The Hales–Jewett theorem states that for any

m

and

r

there exists an

n

such that any

r

-colouring of the elements of

{[m]}^{n}

contains a monochromatic combinatorial line. We study the structure of the wildcard set

S \subseteq [n]

which determines this monochromatic line, showing that when

r

is odd there are

r

-colourings of

{[3]}^{n}

where the wildcard set of a monochromatic line cannot be the union of fewer than

r

intervals. This is tight, as for

n

sufficiently large there are always monochromatic lines whose wildcard set is the union of at most

r

intervals.

Hales-Jewett定理指出，对于任意m和r，存在一个n，使得[m]n的元素的任意r着色包含一条单色组合线。我们研究了决定该单色线的通配符集S≤[n]的结构，证明了当r为奇数时，存在b[3]n的r个着色，其中单色线的通配符集不可能是小于r个区间的并。这是紧密的，因为当n足够大时，总是存在单色线，其通配符集是最多r个区间的并集。

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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 : 2026-02-01 Epub 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

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