European Journal of Combinatorics最新文献

英文中文

Maximizing the number of rational-value sums or zero-sums 使有理值和或零和的数目最大化

IF 0.9 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2025-12-30 DOI: 10.1016/j.ejc.2025.104324

Benjamin Móricz , Zoltán Lóránt Nagy

What is the maximum number of

r

-term sums admitting rational values in

n

-element sets of irrational numbers? We determine the maximum when

r < 4

r \geq n / 2

and also in case when we drop the condition on the number of summands. It turns out that the

r

-term sum problem is equivalent to determine the maximum number of

r

-term zero-sum subsequences in

n

-element sequences of integers, which can be seen as a variant of the famous Erdős–Ginzburg–Ziv theorem.

n元素无理数集合中允许有理数的r项和的最大个数是多少？我们在r<；4或r≥n/2时确定最大值，也在放弃求和个数的条件时确定最大值。事实证明，r项和问题等价于确定n元素整数序列中r项零和子序列的最大个数，这可以看作是著名的Erdős-Ginzburg-Ziv定理的一个变体。

引用次数: 0

On the minimum spanning tree distribution in grids 网格中最小生成树分布的研究

IF 0.9 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2025-12-27 DOI: 10.1016/j.ejc.2025.104325

Kristopher Tapp

We study the minimum spanning tree distribution on the space of spanning trees of the

n

-by-

n

grid for large

n

. We establish bounds on the decay rates of the probability of the most and the least probable spanning trees as

n \to \infty

, and we develop general tools for studying the decay rates of spanning tree families.

我们研究了n × n网格中生成树在n大时的最小生成树分布。我们建立了n→∞时最可能和最小可能生成树的概率衰减率的界，并开发了研究生成树族衰减率的通用工具。

引用次数: 0

Acyclic subgraphs of digraphs with high chromatic number 高色数有向图的无环子图

IF 0.9 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2025-12-26 DOI: 10.1016/j.ejc.2025.104323

Raphael Yuster

For a digraph

G

, let

f (G)

be the maximum chromatic number of an acyclic subgraph of

G

. For an

n

-vertex digraph

G

it is proved that

f (G) \geq n^{5 / 9 - o (1)} s^{- 14 / 9}

where

s

is the bipartite independence number of

G

, i.e., the largest

s

for which there are two disjoint

s

-sets of vertices with no edge between them. This generalizes a result of Fox, Kwan and Sudakov, who proved this for the case

s = 0

(i.e., tournaments and semicomplete digraphs). Consequently, if

s = n^{o (1)}

, then

f (G) \geq n^{5 / 9 - o (1)}

which polynomially improves the folklore bound

f (G) \geq n^{1 / 2 - o (1)}

. As a corollary, with high probability, all orientations of the random

n

-vertex graph with edge probability

p = n^{- o (1)}

(in particular, constant

p

, hence almost all

n

-vertex graphs) satisfy

f (G) \geq n^{5 / 9 - o (1)}

. Our proof uses a theorem of Gallai and Milgram that together with several additional ideas, essentially reduces to the proof of Fox, Kwan and Sudakov.

对于有向图G，设f(G)为G的无环子图的最大色数。对于n顶点有向图G，证明了f(G)≥n5/9−0 (1)s−14/9，其中s为G的二部无关数，即存在两个不相交的无边的顶点s集的最大s。这推广了Fox， Kwan和Sudakov的结果，他们在s=0的情况下证明了这一点（即比赛和半完全有向图）。因此，如果s=no(1)，则f(G)≥n5/9−o(1)，这多项式地改善了民间传说界f(G)≥n1/2−o(1)。作为一个推论，在高概率下，边概率p=n−o(1)的随机n顶点图（特别是常数p，因此几乎所有n顶点图）的所有方向都满足f(G)≥n5/9−o(1)。我们的证明使用了Gallai和Milgram的一个定理，加上一些额外的想法，本质上归结为Fox， Kwan和Sudakov的证明。

引用次数: 0

A self-conjugate partition analog of (t,t+1)-core partitions with distinct parts 具有不同部分的（t,t+1）核划分的自共轭划分模拟

IF 0.9 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2025-12-22 DOI: 10.1016/j.ejc.2025.104322

Huan Xiong, Lihong Yang

Simultaneous core partitions have been extensively studied over the past two decades. In 2013, Amdeberhan proposed several conjectures regarding the number, the average size, and the largest size of

(t, t + 1)

-core partitions with distinct parts. These conjectures were proved and generalized by Straub, Nath-Sellers, Zaleski-Zeilberger, Xiong, Paramonov, and many other mathematicians. In this paper, we introduce a natural self-conjugate partition analog of

(t, t + 1)

-core partitions with distinct parts and derive their number, average size, and largest size.

在过去的二十年里，同步岩心分区得到了广泛的研究。Amdeberhan在2013年提出了关于（t,t+1）个具有不同部分的核分区的数量、平均大小和最大大小的几个猜想。这些猜想被Straub、Nath-Sellers、Zaleski-Zeilberger、Xiong、Paramonov和许多其他数学家证明并推广。本文引入了具有不同部分的（t,t+1）核划分的自然自共轭模拟，并导出了它们的数目、平均大小和最大大小。

引用次数: 0

A new bijective proof of the q-Pfaff–Saalschütz identity with applications to quantum groups q- pfaff - saalsch<e:1>兹恒等式的新双射证明及其在量子群中的应用

IF 0.9 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2025-12-22 DOI: 10.1016/j.ejc.2025.104321

Álvaro Gutiérrez , Álvaro L. Martínez , Michał Szwej , Mark Wildon

We present a combinatorial proof of the

q

-Pfaff–Saalschütz identity by a composition of explicit bijections, in which

q

-binomial coefficients are interpreted as counting subspaces of

F_{q}

-vector spaces. As a corollary, we obtain a new multiplication rule for quantum binomial coefficients and hence a new presentation of Lusztig’s integral form

U_{Z [q, q^{- 1}]} ({sl}_{2})

of the Cartan subalgebra of the quantum group

U_{q} ({sl}_{2})

利用显式双射的复合给出了q- pfaff - saalsch兹恒等式的组合证明，其中q-二项式系数被解释为fq -向量空间的计数子空间。作为推论，我们得到了量子二项式系数的一个新的乘法规则，从而得到了量子群Uq（sl2）的Cartan子代数的Lusztig积分形式UZ[q，q−1]（sl2）的一个新的表示。

引用次数: 0

Restricted chain-order polytopes via combinatorial mutations 限制性链序多构体通过组合突变

IF 0.9 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2025-12-22 DOI: 10.1016/j.ejc.2025.104326

Oliver Clarke , Akihiro Higashitani , Francesca Zaffalon

We study restricted chain-order polytopes associated to Young diagrams using combinatorial mutations. These polytopes are obtained by intersecting chain-order polytopes with certain hyperplanes. The family of chain-order polytopes associated to a poset interpolate between the order and chain polytopes of the poset. Each such polytope retains properties of the order and chain polytope; for example its Ehrhart polynomial. For a fixed Young diagram, we show that all restricted chain-order polytopes are related by a sequence of combinatorial mutations. Since the property of giving rise to the period collapse phenomenon is invariant under combinatorial mutations, we provide a large class of rational polytopes that give rise to period collapse.

利用组合突变研究了与杨氏图相关的限制性链序多面体。这些多面体是由具有一定超平面的链序多面体相交得到的。与偏序集相关的链序多面体族插入在偏序集的序多面体和链多面体之间。每一个这样的多面体都保留了有序多面体和链多面体的性质；比如它的Ehrhart多项式。对于一个固定的Young图，我们证明了所有的限制性链序多面体都是由一系列组合突变联系起来的。由于引起周期坍缩现象的性质在组合突变下是不变的，我们给出了一大类引起周期坍缩的有理多面体。

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 : 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

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 : 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

Slit-slide-sew bijections for constellations and quasiconstellations 星座和准星座的裂隙-滑动-缝双射

IF 0.9 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2025-12-19 DOI: 10.1016/j.ejc.2025.104318

Jérémie Bettinelli , Dimitri Korkotashvili

We extend so-called slit-slide-sew bijections to constellations and quasiconstellations, which allow to recover the counting formula for constellations or quasiconstellations with a given face degree distribution.

More precisely, we present an involution on the set of hypermaps given with an orientation, one distinguished corner, and one distinguished edge leading away from the corner while oriented in the given orientation. This involution reverts the orientation, exchanges the distinguished corner with the distinguished edge in some sense, slightly modifying the degrees of the incident faces in passing, while keeping all the other faces intact.

The construction consists in building a canonical path from the distinguished elements, slitting the map along it, and sewing back after sliding by one unit along the path. The involution specializes into a bijection interpreting combinatorial identities linking the numbers of constellations or quasiconstellations with a given face degree distribution, where the degree distributions differ by one

+ 1

and one

- 1

Our bijections yield a “degree-by-degree, face-by-face” growth algorithm that samples a hypermap uniformly distributed among constellations or quasiconstellations with prescribed face degrees. More precisely, it samples at each step uniform constellations or quasiconstellations, whose face degree distributions slightly evolve to the desired distribution.

我们将所谓的裂缝-滑动-缝双射扩展到星座和准星座，这允许恢复具有给定面度分布的星座或准星座的计数公式。更准确地说，我们给出了一组超映射的对合，这些超映射具有一个方向，一个可分辨的角，以及一个从该角引出的可分辨的边，同时在给定的方向上取向。这种对合恢复了方向，在某种意义上交换了区分角和区分边，稍微修改了经过的事件面的程度，同时保持所有其他面完整。建筑包括从不同的元素建立一个规范的路径，沿着它切割地图，并沿着路径滑动一个单元后缝回。对合专门用于解释组合恒等式的双射，将星座或准星座的数量与给定的面度分布联系起来，其中度分布的差异为1 +1和1 - 1。我们的双射产生了一种“逐度、逐面”的增长算法，该算法对具有规定面度的星座或准星座之间均匀分布的超映射进行采样。更精确地说，它在每一步采样均匀星座或准星座，其面度分布略微进化到所需的分布。

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引用次数: 0

Neighborly boxes and bipartite coverings; constructions and conjectures 相邻的盒子和二部覆盖物；结构和猜想

IF 0.9 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2025-12-18 DOI: 10.1016/j.ejc.2025.104319

Jarosław Grytczuk , Andrzej P. Kisielewicz , Krzysztof Przesławski

Two axis-aligned boxes in

R^{d}

are

k

-neighborly if their intersection has dimension at least

d - k

and at most

d - 1

. The maximum number of pairwise

k

-neighborly boxes in

R^{d}

is denoted by

n (k, d)

. It is known that

n (k, d) = Θ (d^{k})

, for fixed

1 ⩽ k ⩽ d

, but exact formulas are known only in three cases:

k = 1

k = d - 1

, and

k = d

. In particular, the formula

n (1, d) = d + 1

is equivalent to the famous theorem of Graham and Pollak on bipartite partitions of cliques.

In this paper we are dealing with the case

k = 2

. We give a new construction of

k

-neighborly codes giving better lower bounds on

n (2, d)

. The construction is recursive in nature and uses a kind of “algebra” on lists of ternary strings, which encode neighborly boxes in a familiar way. Moreover, we conjecture that our construction is optimal and gives an explicit formula for

n (2, d)

. This supposition is supported by some numerical experiments and some partial results on related open problems which are recalled.

如果两个在Rd中轴对齐的盒子的相交维度至少为d - k且不超过d - 1，则它们是k近邻。Rd中成对k邻框的最大数目用n（k,d）表示。已知n(k,d)=Θ（dk），对于固定的1≤k≤d，但确切的公式只在三种情况下已知：k=1, k=d−1和k=d。特别地，公式n(1,d)=d+1等价于著名的Graham和Pollak关于团的二部分割的定理。在本文中，我们处理k=2的情况。我们给出了一个新的k邻码结构，给出了n（2,d）上更好的下界。这种构造本质上是递归的，并在三元字符串列表上使用了一种“代数”，以一种熟悉的方式对相邻框进行编码。此外，我们推测我们的结构是最优的，并给出了n（2,d）的显式公式。这一假设得到了一些数值实验和相关开放问题的部分结果的支持。

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引用次数: 0

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