European Journal of Combinatorics最新文献

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Necklaces, permutations, and periodic critical orbits for quadratic polynomials 二次多项式的项链，排列和周期临界轨道

IF 0.9 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2026-04-01 Epub Date: 2026-02-12 DOI: 10.1016/j.ejc.2026.104352

Matthew Baker , Andrea Chen , Sophie Li , Matthew Qian

<div><div>Let <math><msub><mrow><mi>G</mi></mrow><mrow><mi>n</mi></mrow></msub></math> denote the <math><mi>n</mi></math>th Gleason polynomial, whose roots correspond to parameters <math><mi>c</mi></math> such that the critical point 0 is periodic of exact period <math><mi>n</mi></math> under iteration of <math><mrow><msup><mrow><mi>z</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mi>c</mi></mrow></math>, and let <math><msub><mrow><mover><mrow><mi>G</mi></mrow><mrow><mo>̄</mo></mrow></mover></mrow><mrow><mi>n</mi></mrow></msub></math> denote the reduction of <math><msub><mrow><mi>G</mi></mrow><mrow><mi>n</mi></mrow></msub></math> modulo 2. Buff, Floyd, Koch, and Parry made the surprising observation that the number of real roots of <math><msub><mrow><mi>G</mi></mrow><mrow><mi>n</mi></mrow></msub></math> is equal to the number of irreducible factors of <math><msub><mrow><mover><mrow><mi>G</mi></mrow><mrow><mo>̄</mo></mrow></mover></mrow><mrow><mi>n</mi></mrow></msub></math> for all <math><mi>n</mi></math>. We provide a bijective proof for this result by first providing explicit bijections between (a) the set of real roots of <math><msub><mrow><mi>G</mi></mrow><mrow><mi>n</mi></mrow></msub></math> and the set <math><mrow><mover><mrow><mi>N</mi></mrow><mrow><mo>̄</mo></mrow></mover><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math> of equivalence classes of primitive binary necklaces of length <math><mi>n</mi></math> under the inversion map swapping 0 and 1; and (b) the set of irreducible factors of <math><msub><mrow><mi>G</mi></mrow><mrow><mi>n</mi></mrow></msub></math> modulo 2 and the set <math><mrow><msup><mrow><mover><mrow><mi>N</mi></mrow><mrow><mo>̃</mo></mrow></mover></mrow><mrow><mo>+</mo></mrow></msup><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math> of binary necklaces which are either primitive of length <math><mi>n</mi></math> with an even number of 1’s or primitive of length <math><mrow><mi>n</mi><mo>/</mo><mn>2</mn></mrow></math> with an odd number of 1’s. We then provide an explicit bijection, closely related to Milnor and Thurston’s kneading theory, between <math><mrow><mover><mrow><mi>N</mi></mrow><mrow><mo>̄</mo></mrow></mover><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math> and <math><mrow><msup><mrow><mover><mrow><mi>N</mi></mrow><mrow><mo>̃</mo></mrow></mover></mrow><mrow><mo>+</mo></mrow></msup><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math>. In addition, we provide explicit bijections between <math><mrow><mover><mrow><mi>N</mi></mrow><mrow><mo>̄</mo></mrow></mover><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math>, the set <math><mrow><mi>CUP</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math>

设Gn表示第n个Gleason多项式，其根对应于参数c，使得临界点0在z2+c迭代下的周期恰好为n，设Ḡn表示Gn模2的约简。Buff, Floyd， Koch和Parry做出了令人惊讶的观察，即对于所有n， Gn的实根的个数等于Ḡn的不可约因子的个数。我们通过首先提供(a) Gn的实根的集合和长度为n的原始二进制项链的等价类的集合n (n)之间的显式双射，为这个结果提供了双射证明；(b) Gn模2的不可约因子集和二进制项链的集合Ñ+(n)，这些项链要么是长度为n的原语，具有偶数个1，要么是长度为n/2的原语，具有奇数个1。然后，我们提供了一个显式双射，与Milnor和Thurston的揉捏理论密切相关，在N (N)和Ñ+(N)之间。此外，我们给出了N (N)、{1，…，N}的循环单峰置换的集合CUP(N)和长度为N且奇数个1的原始二元项链的集合N−(N)之间的显式双射。

引用次数: 0

Strong parity edge-colorings of graphs 图的强奇偶边着色

IF 0.9 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2026-04-01 Epub Date: 2026-01-28 DOI: 10.1016/j.ejc.2026.104343

Peter Bradshaw , Sergey Norin , Douglas B. West

<div><div>An edge-coloring of a graph <math><mi>G</mi></math> assigns a color to each edge of <math><mi>G</mi></math>. An edge-coloring is a parity edge-coloring if for each path <math><mi>P</mi></math> in <math><mi>G</mi></math>, it uses some color on an odd number of edges in <math><mi>P</mi></math>. It is a strong parity edge-coloring if for every open walk <math><mi>W</mi></math> in <math><mi>G</mi></math>, it uses some color an odd number of times along <math><mi>W</mi></math>. The minimum numbers of colors in parity and strong parity edge-colorings of <math><mi>G</mi></math> are denoted <math><mrow><mi>p</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math> and <math><mrow><mover><mrow><mi>p</mi></mrow><mrow><mo>ˆ</mo></mrow></mover><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math>, respectively.</div><div>We characterize strong parity edge-colorings and use this to prove lower bounds on <math><mrow><mover><mrow><mi>p</mi></mrow><mrow><mo>ˆ</mo></mrow></mover><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math> and answer several questions of Bunde, Milans, West, and Wu. The applications are as follows. (1) We prove the conjecture that <math><mrow><mover><mrow><mi>p</mi></mrow><mrow><mo>ˆ</mo></mrow></mover><mrow><mo>(</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow></msub><mo>)</mo></mrow><mo>=</mo><mi>s</mi><mo>∘</mo><mi>t</mi></mrow></math>, where <math><mrow><mi>s</mi><mo>∘</mo><mi>t</mi></mrow></math> is the Hopf–Stiefel function. (2) We show that <math><mrow><mover><mrow><mi>p</mi></mrow><mrow><mo>ˆ</mo></mrow></mover><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math> for a connected <math><mi>n</mi></math>-vertex graph <math><mi>G</mi></math> equals the known lower bound <math><mrow><mo>⌈</mo><msub><mrow><mo>log</mo></mrow><mrow><mn>2</mn></mrow></msub><mi>n</mi><mo>⌉</mo></mrow></math> if and only if <math><mi>G</mi></math> is a subgraph of the hypercube <math><msub><mrow><mi>Q</mi></mrow><mrow><mrow><mo>⌈</mo><msub><mrow><mo>log</mo></mrow><mrow><mn>2</mn></mrow></msub><mi>n</mi><mo>⌉</mo></mrow></mrow></msub></math>. (3) We asymptotically compute <math><mrow><mover><mrow><mi>p</mi></mrow><mrow><mo>ˆ</mo></mrow></mover><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math> when <math><mi>G</mi></math> is the <math><mi>ℓ</mi></math>th distance-power of a path, proving <math><mrow><mover><mrow><mi>p</mi></mrow><mrow><mo>ˆ</mo></mrow></mover><mrow><mo>(</mo><msubsup><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>ℓ</mi></mrow></msubsup><mo>)</mo></mrow><mo>∼</mo><mi>ℓ</mi><mfenced><mrow><msub><mrow><mo>log</mo></mrow><mrow><mn>2</mn></mrow></m

一个图G的edge-coloring分配一个颜色每条边的G . edge-coloring是平价edge-coloring如果每个路径P G,它使用一些颜色在奇数边P .这是一个强大的平价edge-coloring如果每打开走W G,它使用一些颜色奇数倍的W .平价的最低数量的颜色和强劲的平价edge-colorings G P (G)和P表示ˆ(G),分别。我们刻画了强奇偶边着色，并用它证明了p ^ (G)的下界，并回答了Bunde、Milans、West和Wu的几个问题。应用如下：(1)证明p´(Ks,t)=s°t的猜想，其中s°t是Hopf-Stiefel函数。(2)我们证明了连通n顶点图G的p ^ (G)等于已知的下界≤lg2n≤当且仅当G是超立方体Q≤lg2n≤的子图。(3)当G为路径的距离幂的第n次幂时，我们渐近地计算p ^ (G)，证明了p ^ (Pn) ~ （log2n）。(4)通过构造二部图G使p φ (G)/p(G)任意大，证明了当G是二部图时p φ (G)=p(G)的猜想；特别是p(G)≥1−0 (1)3klnk, p(G)≤2k+k1/3。

引用次数: 0

A weighted Murnaghan–Nakayama rule for (P,ω)-partitions （P,ω）-分区的加权Murnaghan-Nakayama规则

IF 0.9 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2026-04-01 Epub Date: 2026-01-12 DOI: 10.1016/j.ejc.2025.104332

Per Alexandersson , Olivia Nabawanda

The

(P, ω)

-partition generating function

K_{(P, ω)} (x)

is a quasisymmetric function obtained from a labeled poset. Recently, Liu and Weselcouch gave a formula for the coefficients of

K_{(P, ω)} (x)

when expanded in the quasisymmetric power sum function basis. This formula generalizes the classical Murnaghan–Nakayama rule for Schur functions.

We extend this result to weighted

(P, ω)

-partitions and provide a short combinatorial proof, avoiding the Hopf algebra machinery used by Liu–Weselcouch.

（P,ω）分块生成函数K(P,ω)(x)是由标记偏序集得到的拟对称函数。最近，Liu和Weselcouch给出了K(P,ω)(x)在拟对称幂和函数基上展开时的系数公式。这个公式推广了Schur函数的经典Murnaghan-Nakayama规则。我们将这个结果推广到加权（P,ω）分区，并提供了一个简短的组合证明，避免了Liu-Weselcouch使用的Hopf代数机制。

引用次数: 0

Majority bootstrap percolation on the permutahedron and other high-dimensional graphs 在多面体和其他高维图上的大多数自举渗流

IF 0.9 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2026-04-01 Epub Date: 2026-02-04 DOI: 10.1016/j.ejc.2026.104347

Maurício Collares , Joshua Erde , Anna Geisler , Mihyun Kang

Majority bootstrap percolation is a model of infection spreading in networks. Starting with a set of initially infected vertices, new vertices become infected once half of their neighbours are infected. Balogh, Bollobás and Morris studied this process on the hypercube and showed that there is a phase transition as the density of the initially infected set increases. Generalising their results to a broad class of high-dimensional graphs, the authors of this work established similar bounds on the critical window, establishing a universal behaviour for these graphs.

These methods necessitated an exponential bound on the order of the graphs in terms of their degrees. In this paper, we consider a slightly more restrictive class of high-dimensional graphs, which nevertheless covers most examples considered previously. Under these stronger assumptions, we are able to show that this universal behaviour holds in graphs of superexponential order. As a concrete and motivating example, we apply this result to the permutahedron, a symmetric high-dimensional graph of superexponential order which arises naturally in many areas of mathematics. Our methods also allow us to slightly improve the bounds on the critical window given in previous work, in particular in the case of the hypercube.

Finally, the upper and lower bounds on the critical window depend on the maximum and minimum degree of the graph, respectively, leading to much worse bounds for irregular graphs. We also analyse an explicit example of a high-dimensional irregular graph, namely the Cartesian product of stars and determine the first two terms in the expansion of the critical probability, which in this case is determined by the minimum degree.

多数自举渗透是一种网络感染传播模型。从一组最初被感染的顶点开始，一旦新顶点的一半邻居被感染，它们就会被感染。Balogh， Bollobás和Morris在超立方体上研究了这一过程，并表明随着初始感染集密度的增加，存在一个相变。将他们的结果推广到广泛的高维图中，作者在临界窗口上建立了类似的边界，建立了这些图的普遍行为。这些方法需要在图的度数顺序上有一个指数界。在这篇论文中，我们考虑了一种稍微有限制的高维图，它涵盖了之前考虑的大多数例子。在这些更强的假设下，我们能够证明这种普遍行为在超指数阶的图中成立。作为一个具体的和激励的例子，我们将这个结果应用于复面体，一个在数学的许多领域中自然出现的超指数阶的对称高维图。我们的方法还允许我们稍微改进先前工作中给出的临界窗口的边界，特别是在超立方体的情况下。最后，临界窗口的上界和下界分别取决于图的最大和最小度，导致不规则图的边界更差。我们还分析了一个高维不规则图的显式例子，即恒星的笛卡尔积，并确定了临界概率展开式中的前两项，在这种情况下，临界概率由最小度决定。

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引用次数: 0

On the anti-Ramsey threshold 在反拉姆齐的门槛上

IF 0.9 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2026-04-01 Epub Date: 2026-01-22 DOI: 10.1016/j.ejc.2026.104344

Eden Kuperwasser

We say that a graph

G

is anti-Ramsey for a graph

H

if any proper edge-colouring of

G

yields a rainbow copy of

H

, i.e. a copy of

H

whose edges all receive different colours. In this work we determine the threshold at which the binomial random graph becomes anti-Ramsey for any fixed graph

H

, given that

H

is sufficiently dense. Our proof employs a graph decomposition lemma in the style of the Nine Dragon Tree theorem, which may be of independent interest.

我们说图G对于图H是反拉姆齐的，如果G的任何适当的边着色产生H的彩虹副本，即H的一个副本，其所有的边都得到不同的颜色。在这个工作中，我们确定了二项随机图成为任意固定图H的反拉姆齐的阈值，假设H足够密集。我们的证明采用了九龙树定理风格的图分解引理，这可能是独立的兴趣。

引用次数: 0

Perfect matroids over skew hyperfields 倾斜超场上的完美拟阵

IF 0.9 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2026-04-01 Epub Date: 2026-01-09 DOI: 10.1016/j.ejc.2025.104327

Nathan Bowler , Rudi Pendavingh

A hyperfield

H

is stringent if

a ⊞ b

is a singleton unless

a = - b

, for all

a, b \in H

. By a construction of Marc Krasner, each valued field gives rise to a stringent hyperfield. We show that if

H

is a stringent skew hyperfield, then weak matroids over

H

are strong matroids over

H

. Also, we present vector axioms for matroids over stringent skew hyperfields which generalize the vector axioms for oriented matroids and valuated matroids.

对于所有的A，b∈H，如果A + b是单态，则超场H是严格的，除非A = - b。通过Marc Krasner的构造，每个有值场都会产生一个严格的超场。我们证明了如果H是一个严格的斜超场，那么H上的弱拟阵就是H上的强拟阵。同时，我们给出了严格斜超场上的拟阵的向量公理，推广了有向拟阵和赋值拟阵的向量公理。

引用次数: 0

Convergence of spectra of digraph limits 有向图极限谱的收敛性

IF 0.9 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2026-04-01 Epub Date: 2026-02-09 DOI: 10.1016/j.ejc.2026.104349

Jan Grebík , Daniel Král’ , Xizhi Liu , Oleg Pikhurko , Julia Slipantschuk

The relation between densities of cycles and the spectrum of a graphon, which implies that the spectra of convergent graphons converge, fundamentally relies on the self-adjointness of the linear operator associated with a graphon. In this short paper, we consider the setting of digraphons, which are limits of directed graphs, and prove that the spectra of convergent digraphons converge. Using this result, we establish the relation between densities of directed cycles and the spectrum of a digraphon.

循环密度与石墨子谱之间的关系从根本上依赖于与石墨子相关的线性算子的自伴随性，这意味着收敛石墨子的谱是收敛的。在这篇简短的文章中，我们考虑有向图的极限——有向图的集合，并证明收敛有向图的谱是收敛的。利用这一结果，我们建立了有向环密度与有向图子谱之间的关系。

引用次数: 0

Recovery of cyclic words by their subwords 根据循环词的子词恢复循环词

IF 0.9 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2026-04-01 Epub Date: 2026-01-02 DOI: 10.1016/j.ejc.2025.104329

Sergey Luchinin , Svetlana Puzynina , Michaël Rao

The problem of reconstructing words from their subwords involves determining the minimum amount of information needed, such as multisets of scattered subwords of a specific length or the frequency of scattered subwords from a given set, in order to uniquely identify a word. In this paper we show that a cyclic word on a binary alphabet can be reconstructed by its scattered subwords of length

\frac{3}{4} n + 4

, and for each

n

one can find two cyclic words of length

n

which have the same set of scattered subwords of length

\frac{3}{4} n - \frac{3}{2}

从词的子词重建词的问题涉及确定所需的最小信息量，例如特定长度的分散子词的多集或来自给定集的分散子词的频率，以便唯一地标识一个词。本文证明了二进制字母表上的一个循环字可以用它的长度为34n+4的分散子字来重构，并且对于每一个n，可以找到两个长度为n的循环字，它们具有相同的长度为34n−32的分散子字集。

引用次数: 0

Product representation of perfect cubes 完全立方的积表示

IF 0.9 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2026-04-01 Epub Date: 2026-01-23 DOI: 10.1016/j.ejc.2026.104342

Zsigmond György Fleiner , Márk Hunor Juhász , Blanka Kövér , Péter Pál Pach , Csaba Sándor

Let

F_{k, d} (n)

be the maximal size of a set

A \subseteq [n]

such that the equation

a_{1} a_{2} \dots a_{k} = x^{d}, a_{1} < a_{2} < \dots < a_{k}

has no solution with

a_{1}, a_{2}, \dots, a_{k} \in A

and integer

x

. Erdős, Sárközy and T. Sós studied

F_{k, 2}

, and gave bounds when

k = 2, 3, 4, 6

and also in the general case. We study the problem for

d = 3

, and provide bounds for

k = 2, 3, 4, 6

and 9, as well as in the general case. In particular, we refute an 18-year-old conjecture of Verstraëte.

We also introduce another function

f_{k, d}

closely related to

F_{k, d}

: While the original problem requires

a_{1}, \dots, a_{k}

to all be distinct, we can relax this and only require that the multiset of the

a_{i}

’s cannot be partitioned into

d

-tuples where each

d

-tuple consists of

d

copies of the same number.

设Fk，d(n)为集合a的最大大小，使得方程a1a2⋯ak=xd,a1<a2<⋯<； ak对a1，a2，…，ak∈a和整数x无解。Erdős, Sárközy， T. Sós研究了Fk，2，并给出了k=2,3,4,6及一般情况下的界。我们研究了d=3时的问题，并给出了k=2、3、4、6、9以及一般情况下的界。特别是，我们反驳了一个18年的猜想Verstraëte。我们还引入另一个与fk，d密切相关的函数fk，d：虽然原始问题要求a1，…，ak都是不同的，但我们可以放宽这一点，只要求ai的多集不能划分为d元组，其中每个d元组由相同数量的d个副本组成。

引用次数: 0

The partial derivative of ratios of Schur polynomials and applications to symplectic quotients 舒尔多项式比值的偏导数及其在辛商中的应用

IF 0.9 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2026-04-01 Epub Date: 2026-02-07 DOI: 10.1016/j.ejc.2026.104351

Hans-Christian Herbig , Daniel Herden , Harper Kolehmainen , Christopher Seaton

We show that a ratio of Schur polynomials

s_{λ} / s_{ρ}

associated to partitions

λ

and

ρ

such that

λ ⊊ ρ

has a negative partial derivative at any point where all variables are positive. This is accomplished by establishing an injective map between sets of pairs of skew semistandard Young tableaux that preserves the product of the corresponding monomials. We use this result and the description of the first Laurent coefficient of the Hilbert series of the graded algebra of regular functions on a linear symplectic quotient by the circle to demonstrate that many such symplectic quotients are not graded regularly diffeomorphic. In addition, we give an upper bound for this Laurent coefficient in terms of the largest two weights of the circle representation and demonstrate that all but finitely many circle symplectic quotients of each dimension are not graded regularly diffeomorphic to linear symplectic quotients by

{SU}_{2}

我们证明了与分区λ和ρ相关的舒尔多项式的比值λ/ sp使得λ≠ρ在所有变量为正的任何点上具有负偏导数。这是通过在一组歪斜半标准杨表对之间建立一个内射映射来实现的，该映射保留了相应单项式的乘积。利用这一结果和线性辛商上正则函数的梯度代数的Hilbert级数的第一Laurent系数的圆描述，证明了许多这样的辛商不是梯度正则微分同态的。此外，我们用圆表示的最大两个权值给出了这个洛朗系数的上界，并证明了除了有限个圆外，每个维的所有圆辛商都不是用SU2正则微分同构到线性辛商的。

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