Journal of Combinatorial Theory Series A最新文献

英文中文

Extremal eigenvalues with respect to graph minors 关于图副的极值特征值

IF 1.2 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2026-02-03 DOI: 10.1016/j.jcta.2026.106164

Mingqing Zhai , Longfei Fang , Huiqiu Lin

<div><div>Minors play a crucial role in various branches of graph theory, including structural graph theory, extremal graph theory, and topological graph theory, and have garnered significant interest in these areas. This paper explores the maximal spectral radius, denoted <math><mi>s</mi><mi>p</mi><mi>e</mi><mi>x</mi><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>H</mi></mrow><mrow><mi>m</mi><mi>i</mi><mi>n</mi><mi>o</mi><mi>r</mi></mrow></msub><mo>)</mo></math>, of n-vertex graphs that exclude any graph from a fixed family <math><mi>H</mi></math> as a minor.</div><div>We derive the asymptotic value for <math><mi>s</mi><mi>p</mi><mi>e</mi><mi>x</mi><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>H</mi></mrow><mrow><mi>m</mi><mi>i</mi><mi>n</mi><mi>o</mi><mi>r</mi></mrow></msub><mo>)</mo></math> and establish a unified stable structure for extremal graphs by introducing a novel application of the absorbing method to eigenvalue analysis, along with models and partitions with respect to a general H minor. In particular, we prove three central theorems, the most fundamental of which asserts that every graph with spectral radius <math><mi>ρ</mi><mo>≥</mo><mi>s</mi><mi>p</mi><mi>e</mi><mi>x</mi><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mo>{</mo><mi>H</mi><mo>}</mo></mrow><mrow><mi>m</mi><mi>i</mi><mi>n</mi><mi>o</mi><mi>r</mi></mrow></msub><mo>)</mo></math> contains either an H minor or a spanning book <math><msub><mrow><mi>B</mi></mrow><mrow><msub><mrow><mi>γ</mi></mrow><mrow><mi>H</mi></mrow></msub><mo>,</mo><mi>n</mi><mo>−</mo><msub><mrow><mi>γ</mi></mrow><mrow><mi>H</mi></mrow></msub></mrow></msub></math>, where <math><msub><mrow><mi>γ</mi></mrow><mrow><mi>H</mi></mrow></msub><mo>=</mo><mo>|</mo><mi>H</mi><mo>|</mo><mo>−</mo><msub><mrow><mi>α</mi></mrow><mrow><mi>H</mi></mrow></msub><mo>−</mo><mn>1</mn></math> and <math><msub><mrow><mi>α</mi></mrow><mrow><mi>H</mi></mrow></msub></math> is the independence number of H.</div><div>These three theorems, combined with detailed combinatorial analysis, enable us to determine <math><mi>s</mi><mi>p</mi><mi>e</mi><mi>x</mi><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mo>{</mo><mi>H</mi><mo>}</mo></mrow><mrow><mi>m</mi><mi>i</mi><mi>n</mi><mi>o</mi><mi>r</mi></mrow></msub><mo>)</mo></math> for every complete r-partite graph H. This extends the result of Tait for <math><mi>s</mi><mi>p</mi><mi>e</mi><mi>x</mi><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mo>{</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>}</mo></mrow><mrow><mi>m</mi><mi>i</mi><mi>n</mi><mi>o</mi><mi>r</mi></mrow></msub><mo>)</mo></math> and provides a stronger solution to his conjecture for <math><mi>s</mi><mi>p</mi><mi>e</mi><mi>x</mi><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mo>{</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mro

未成年人在图论的各个分支中扮演着至关重要的角色，包括结构图论、极值图论和拓扑图论，并在这些领域获得了极大的兴趣。本文研究了n顶点图的最大谱半径，记为spex(n,Hminor)，它排除了固定族H中的任何图作为子图。我们推导了spex（n,Hminor）的渐近值，并通过引入吸收方法在特征值分析中的新应用，以及关于一般Hminor的模型和划分，建立了极值图的统一稳定结构。特别地，我们证明了三个中心定理，其中最基本的定理证明了每一个谱半径ρ≥spex（n，{H}次）的图都包含一个H次或一个生成书BγH，n−γH，其中γH=|H|−αH−1和αH是H的独立数。这扩展了Tait关于spex（n,{Kr}minor）的结果，并为他关于spex（n,{Ks,t}minor）的猜想提供了一个更强的解[J]。Combin。Ser的理论。[j].农业科学学报，2019。此外，这些定理暗示或强化了其他已有的关于次元的特征值极值结果，如Tait和Tobin的平方图，Chen、Liu和Zhang的Kr−E(H)次元，以及He、Li和Feng的友谊图次元。

引用次数: 0

A characterization of tetravalent half-arc-transitive graphs of girth 5 周长为5的四价半弧传递图的表征

IF 1.2 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2026-01-29 DOI: 10.1016/j.jcta.2026.106167

Primož Šparl , Jin-Xin Zhou

A graph is said to be half-arc-transitive if its automorphism group is transitive on the vertices and the edges of the graph but not on its arcs. Tetravalent half-arc-transitive graphs of girth 3 were characterized by Marušič and Xu in 1997, while those of girth 4 were characterized by Marušič and Nedela in 2002 and by Potočnik and Wilson in 2007. The investigation of tetravalent half-arc-transitive graphs of girth 5 was initiated by Antončič and Šparl in 2023. In this paper, a characterization of all tetravalent half-arc-transitive graphs of girth 5 is given, and as an application, two open questions from Antončič and Šparl (2023) [1] are answered.

如果图的自同构群在图的顶点和边上可传递，但在图的弧上不能传递，则称图是半弧可传递的。周长3的四价半弧传递图由Marušič和Xu（1997）表示，周长4的四价半弧传递图由Marušič和Nedela（2002）表示，poto nik和Wilson（2007）表示。周长为5的四价半弧传递图的研究是由antoniizi和Šparl于2023年发起的。本文给出了周长为5的所有四价半弧传递图的一个性质，并作为应用，回答了antoniizi和Šparl(2023)[1]提出的两个开放性问题。

引用次数: 0

Solutions to some problems on unique representation bases 在独特的代表性基础上解决一些问题

IF 1.2 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2026-01-29 DOI: 10.1016/j.jcta.2026.106166

Yuchen Ding

In this note, three 2003 problems of Nathanson and two 2007 problems of Chen on unique representation bases for the integers are resolved.

本文解决了2003年Nathanson的三个问题和2007年Chen的两个关于整数唯一表示基的问题。

引用次数: 0

Corrigendum to “Quantum nilpotent subalgebras of classical quantum groups and affine crystals” [J. Comb. Theory, Ser. A 168 (2019) 219–254] “经典量子群和仿射晶体的量子幂零子代数”的更正[J]。合成杆。理论,爵士。[A] [168 (2019) 219-254]

IF 1.2 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2026-01-27 DOI: 10.1016/j.jcta.2026.106161

Il-Seung Jang , Jae-Hoon Kwon

In this corrigendum, we revise [1, Lemma 5.12] and give a revised proof of [1, Lemma 5.14(2)].

在这个勘误表中，我们修正了[1，引理5.12]，并给出了[1，引理5.14(2)]的一个修正证明。

引用次数: 0

On a d-degree Erdős–Ko–Rado Theorem 关于d次Erdős-Ko-Rado定理

IF 1.2 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2026-01-26 DOI: 10.1016/j.jcta.2026.106163

Hao Huang , Yi Zhang

A family of subsets

F

is intersecting if

A \cap B \neq \emptyset

for any

A, B \in F

. In this paper, we show that for given integers

k > d \geq 2

and

n \geq 2 k + 2 d - 3

, and any intersecting family

F

of k-subsets of

{1, \dots, n}

, there exists a d-subset of

[n]

contained in at most

(\begin{matrix} n - d - 1 \\ k - d - 1 \end{matrix})

subsets of

F

. This result, proved using spectral graph theory, gives a d-degree generalization of the celebrated Erdős–Ko–Rado Theorem, improving a theorem of Kupavskii.

对于任意A，B∈F，如果A∩B≠∅，则子集F相交。在本文中，我们证明了对于给定的整数k>；d≥2和n≥2k+2d−3，以及{1，⋯，n}的k个子集的任何相交族F，存在一个d-子集[n]包含在F的最多（n−d−1k−d−1）个子集中。这个结果，用谱图理论证明，给出了著名的Erdős-Ko-Rado定理的d度推广，改进了Kupavskii的一个定理。

引用次数: 0

On r-Euler-Mahonian statistics for multipermutations 关于多置换的r-Euler-Mahonian统计量

IF 1.2 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2026-01-23 DOI: 10.1016/j.jcta.2026.106165

Kaimei Huang , Zhicong Lin , Sherry H.F. Yan

A pair

({st}_{1}, {st}_{2})

of permutation statistics is said to be r-Euler-Mahonian over multipermutations if

({st}_{1}, {st}_{2})

and

(r des

r maj)

are equidistributed over the set

S_{M}

of all multipermutations of M for any given multiset M, where

r des

denotes the r-descent number and

r maj

denotes the r-major index introduced by Rawlings. In this paper, we shall introduce the r-gap excedance number

r exc

and the r-gap Denert's statistic

r den

for multipermutations and prove that

(r exc, r den)

is r-Euler-Mahonian over multipermutations, thereby extending Liu's result on permutations to multipermutations. When

r = 1

, our result recovers the equidistribution of

(des, maj)

and

(exc, den)

over

S_{M}

derived by Han.

对于任意给定的多集合M，如果（st1,st2）和（rdes, rmaj）在M的所有多置换的集合SM上是等分布的，我们称一对（st1,st2）为多置换上的r-Euler-Mahonian，其中rdes表示r-下降数，rmaj表示罗林斯引入的r-主指标。本文引入r-gap超越数rexc和r-gap Denert的多置换统计量rden，并证明（rexc,rden）在多置换上是r-Euler-Mahonian，从而将Liu关于置换的结果推广到多置换。当r=1时，我们的结果恢复了Han导出的（des,maj）和（exc,den）在SM上的均匀分布。

引用次数: 0

Domino tilings, nonintersecting lattice paths and subclasses of Koutschan–Krattenthaler–Schlosser determinants 多米诺平铺、不相交点阵路径和Koutschan-Krattenthaler-Schlosser行列式的子类

IF 1.2 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2026-01-22 DOI: 10.1016/j.jcta.2026.106159

Qipin Chen , Shane Chern , Atsuro Yoshida

Koutschan, Krattenthaler and Schlosser recently considered a family of binomial determinants. In this work, we give combinatorial interpretations of two subclasses of these determinants in terms of domino tilings and nonintersecting lattice paths, thereby partially answering a question of theirs. Furthermore, the determinant evaluations established by Koutschan, Krattenthaler and Schlosser produce many product formulas for our weighted enumerations of domino tilings and nonintersecting lattice paths. However, there are still two enumerations left corresponding to conjectural formulas made by the three. We hereby prove the two conjectures using the principle of holonomic Ansatz plus the approach of modular reduction for creative telescoping, and hence fill the gap.

Koutschan， Krattenthaler和Schlosser最近研究了一类二项式决定因素。在这项工作中，我们给出了这些行列式的两个子类在多米诺骨牌平铺和不相交的晶格路径方面的组合解释，从而部分地回答了他们的问题。此外，由Koutschan， Krattenthaler和Schlosser建立的行列式评估为我们的多米诺骨牌瓦片和非相交晶格路径的加权枚举产生了许多乘积公式。然而，仍然有两个枚举与三人的猜想公式相对应。本文利用完整的Ansatz原理和创造性伸缩的模约化方法证明了这两个猜想，从而填补了这一空白。

引用次数: 0

Van Lint–MacWilliams' conjecture and maximum cliques in Cayley graphs over finite fields, II 有限域上Cayley图的Van Lint-MacWilliams猜想和最大团，2

IF 1.2 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2026-01-19 DOI: 10.1016/j.jcta.2026.106162

Chi Hoi Yip

The well-known Van Lint–MacWilliams' conjecture states that if q is an odd prime power, and

A \subseteq F_{q^{2}}

such that

0, 1 \in A

| A | = q

, and

a - b

is a square for each

a, b \in A

, then A must be the subfield

F_{q}

. This conjecture was first proved by Blokhuis and is often phrased in terms of the maximum cliques in Paley graphs of square order. Previously, Asgarli and the author extended Blokhuis' theorem to a larger family of Cayley graphs. In this paper, we give a new simple proof of Blokhuis' theorem and its extensions. More generally, we show that if

S \subseteq F_{q^{2}}^{⁎}

has small multiplicative doubling, and

A \subseteq F_{q^{2}}

with

0, 1 \in A

| A | = q

, such that

A - A \subseteq S \cup {0}

, then

A = F_{q}

. This new result refines and extends several previous works; moreover, our new approach avoids using heavy machinery from number theory.

著名的Van Lint-MacWilliams猜想认为，如果q是奇质数幂，且A∈Fq2使得0,1∈A, |A|=q，且A - b是每个A， b∈A的平方，则A一定是子域Fq。这个猜想最早是由Blokhuis证明的，并且通常用平方阶Paley图中的最大团来表述。先前，Asgarli和作者将Blokhuis定理推广到更大的Cayley图族。本文给出了Blokhuis定理的一个新的简单证明及其扩展。更一般地说，我们证明了如果S≥≥1，且A≥≥1∈A， |≤A≤|=q，则A≤Fq。这个新结果完善和扩展了以前的几项工作；此外，我们的新方法避免了使用数论中的重型机器。

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引用次数: 0

Helly numbers for quantitative Helly-type results Helly数用于定量的Helly型结果

IF 1.2 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2026-01-16 DOI: 10.1016/j.jcta.2026.106160

Grigory Ivanov , Márton Naszódi

We obtain three Helly-type results. First, we establish a Quantitative Colorful Helly-type theorem with the optimal Helly number 2d concerning the diameter of the intersection of a family of convex bodies. Second, we prove a Quantitative Helly-type theorem with the optimal Helly number

2 d + 1

for the pointwise minimum of logarithmically concave functions. Finally, we present a colorful version of the latter result with Helly number (number of color classes)

3 d + 1

; however, we have no reason to believe that this bound is sharp.

我们得到了三个helly型结果。首先，我们建立了关于一类凸体交点直径的最优Helly数2d的彩色Helly型定量定理。其次，我们证明了对数凹函数的点最小值的最优Helly数2d+1的定量Helly型定理。最后，我们给出了后一种结果的彩色版本，Helly数（颜色类数）3d+1；然而，我们没有理由相信这个界限是尖锐的。

引用次数: 0

Partitions with Durfee triangles of fixed size 具有固定大小的Durfee三角形的分区

IF 1.2 2区数学 Q2 MATHEMATICS

Journal of Combinatorial Theory Series A

Pub Date : 2026-01-12 DOI: 10.1016/j.jcta.2026.106158

N. Guru Sharan , Armin Straub

A well-studied statistic of an integer partition is the size of its Durfee square. In particular, the number

D_{k} (n)

of partitions of n with Durfee square of fixed size k has a well-known simple rational generating function. We study the number

R_{k} (n)

of partitions of n with Durfee triangle of size k (the largest subpartition with parts

1, 2, \dots, k

). We determine the corresponding generating functions which are rational functions of a similar form. Moreover, we explicitly determine the leading asymptotic of

R_{k} (n)

, as

n \to \infty

整数分区的一个被充分研究的统计数据是它的Durfee平方的大小。特别是，具有固定大小k的Durfee平方的n分区的个数Dk(n)具有众所周知的简单有理生成函数。我们研究了大小为k的Durfee三角形（包含1,2，…，k部分的最大子划分）的n分区的个数Rk(n)。我们确定了相应的生成函数，它们是形式相似的有理函数。此外，我们明确地确定了Rk(n)的前渐近，即n→∞。

引用次数: 0

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