European Journal of Combinatorics最新文献

英文中文

Graph classes with few P4’s: Universality and Brownian graphon limits 少P4的图类：普适性和布朗图极限

IF 0.9 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2026-01-30 DOI: 10.1016/j.ejc.2026.104345

Théo Lenoir

We consider large uniform labeled random graphs in different classes with few induced

P_{4}

(

P_{4}

is the graph consisting of a single line of 4 vertices) which generalize the case of cographs. Our main result is the convergence to a Brownian limit object in the space of graphons. As a by-product we obtain new asymptotic enumerative results for all these graph classes. We also obtain typical density results for a wide variety of labeled induced subgraphs. These asymptotics hold at a smaller scale than what is observable through the graphon convergence.

Our proofs rely on tree encoding of graphs. We then use mainly combinatorial arguments, including the symbolic method and singularity analysis.

我们考虑具有少量诱导P4 （P4是由4个顶点的单线组成的图）的不同类别的大型均匀标记随机图，它推广了图的情况。我们的主要结果是收敛到图形空间中的布朗极限对象。作为一个副产品，我们得到了所有这些图类的新的渐近枚举结果。我们还获得了各种标记诱导子图的典型密度结果。这些渐近性在比通过石墨收敛观察到的更小的尺度上成立。我们的证明依赖于图的树编码。然后我们主要使用组合论证，包括符号方法和奇点分析。

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 : 2026-01-28 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

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 : 2026-01-23 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

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 : 2026-01-22 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

Strong modeling limits of graphs with bounded tree-width 有界树宽图的强建模限制

IF 0.9 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2026-01-19 DOI: 10.1016/j.ejc.2025.104330

Andrzej Grzesik , Daniel Král , Samuel Mohr

The notion of first order convergence of graphs unifies the notions of convergence for sparse and dense graphs. Nešetřil and Ossona de Mendez (2019) proved that every first order convergent sequence of graphs from a nowhere-dense class of graphs has a modeling limit and conjectured the existence of such modeling limits with an additional property, the strong finitary mass transport principle. The existence of modeling limits satisfying the strong finitary mass transport principle was proved for first order convergent sequences of trees by Nešetřil and Ossona de Mendez (2016) and for first order sequences of graphs with bounded path-width by Gajarský et al. (2017). We establish the existence of modeling limits satisfying the strong finitary mass transport principle for first order convergent sequences of graphs with bounded tree-width.

图的一阶收敛性的概念统一了稀疏图和密集图的收敛性的概念。Nešetřil和Ossona de Mendez（2019）证明了来自无密度图类的每一个一阶收敛图序列都有一个建模极限，并通过一个附加性质，即强有限质量输运原理，推测了这种建模极限的存在。Nešetřil和Ossona de Mendez（2016）证明了树的一阶收敛序列满足强有限质量输运原理的建模极限的存在性，Gajarský等人（2017）证明了路径宽度有界的图的一阶序列满足强有限质量输运原理。建立了树宽有界的一阶收敛图序列满足强有限质量输运原理的建模极限的存在性。

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 : 2026-01-12 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

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 : 2026-01-09 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

Single-element extensions of matroids over skew tracts 斜束上拟阵的单元扩展

IF 0.9 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2026-01-05 DOI: 10.1016/j.ejc.2025.104328

Ting Su

Matroids over skew tracts provide an algebraic framework simultaneously generalizing the notions of linear subspaces, matroids, oriented matroids, phased matroids, and some other “matroids with extra structure”. A single-element extension of a matroid

M

over a skew tract

T

is a matroid

\tilde{M}

over

T

obtained from

M

by adding one more element. Crapo characterized single-element extensions of ordinary matroids, and Las Vergnas characterized single-element extensions of oriented matroids, in terms of single-element extensions of their rank 2 contractions. The results of Crapo and Las Vergnas do not generalize to matroids over skew tracts, but we will show a necessary and sufficient condition on skew tracts, called Pathetic Cancellation, such that the result can generalize to weak matroids over skew tracts.

Stringent skew hyperfields are a special case of skew tracts which behave in many ways like skew fields. We find a characterization of single-element extensions of strong matroids over stringent skew hyperfields.

斜束上的拟阵提供了一个代数框架，同时推广了线性子空间、拟阵、定向拟阵、相拟阵和其他一些“带额外结构的拟阵”的概念。斜束T上的矩阵M的单元素扩展是由M再加一个元素得到的矩阵M ~ / T。Crapo用普通拟阵的单元扩展来表示普通拟阵，Las Vergnas用定向拟阵的2阶缩缩的单元扩展来表示定向拟阵的单元扩展。Crapo和Las Vergnas的结果不能推广到偏束上的拟阵，但我们将给出一个关于偏束的充要条件，称为可悲抵消，使得结果可以推广到偏束上的弱拟阵。严格倾斜超场是倾斜束的一种特殊情况，它在许多方面表现得像倾斜场。我们得到了强拟阵在严格偏超场上的单元扩展的一个刻划。

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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 : 2026-01-02 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

A note on the maximum diversity of intersecting families in the symmetric group 对称群中相交族的最大多样性的注记

IF 0.9 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2025-12-31 DOI: 10.1016/j.ejc.2025.104331

Jian Wang , Jimeng Xiao

Let

S_{n}

be the symmetric group on the set

[n] ≔ {1, 2, \dots, n}

. A family

F \subset S_{n}

is called intersecting if for every

σ, π \in F

there exists some

i \in [n]

such that

σ (i) = π (i)

. Deza and Frankl proved that the largest intersecting family of permutations is the full star, that is, the collection of all permutations with a fixed position. The diversity of an intersecting family

F

is defined as the minimum number of permutations in

F

, whose deletion results in a star. In the present paper, by applying the spread approximation method developed recently by Kupavskii and Zakharov, we prove that for

n \geq 500

the diversity of an intersecting subfamily of

S_{n}

is at most

(n - 3) (n - 3)!

, which is best possible.

设Sn是集合[n]上的对称群，其中包括{1,2，…，n}。如果对于每一个σ，π∈F存在某个i∈[n]使得σ(i)=π(i)，则称族F∧Sn相交。Deza和Frankl证明了最大的相交排列族是全星形排列，即所有位置固定的排列的集合。交叉家族F的多样性定义为F中最小排列数，其缺失导致星形。本文利用Kupavskii和Zakharov最近提出的扩展逼近方法，证明了当n≥500时，Sn的相交子族的分集不超过（n−3）（n−3）！这是最好的选择。

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引用次数: 0

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European Journal of Combinatorics

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