European Journal of Combinatorics最新文献

英文中文

On the order of semiregular automorphisms of cubic vertex-transitive graphs 论立方顶点变换图的半圆自动形的阶数

IF 1 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2024-11-26 DOI: 10.1016/j.ejc.2024.104091

Marco Barbieri , Valentina Grazian , Pablo Spiga

We prove that, if

Γ

is a finite connected cubic vertex-transitive graph, then either there exists a semiregular automorphism of

Γ

of order at least 6, or the number of vertices of

Γ

is bounded above by an absolute constant.

我们证明，如果 Γ 是一个有限连接的立方顶点传递图，那么要么存在阶数至少为 6 的 Γ 的半圆自动形，要么 Γ 的顶点数在上面以一个绝对常数为界。

引用次数: 0

More on rainbow cliques in edge-colored graphs 边色图中彩虹小群的更多内容

IF 1 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2024-11-25 DOI: 10.1016/j.ejc.2024.104088

Xiao-Chuan Liu , Danni Peng , Xu Yang

In an edge-colored graph

G

, a rainbow clique

K_{k}

is a complete subgraph on

k

vertices in which all the edges have distinct colors. Let

e (G)

and

c (G)

be the number of edges and colors in

G

, respectively. In this paper, we show that for any

ɛ > 0

, if

e (G) + c (G) \geq (1 + \frac{k - 3}{k - 2} + 2 ɛ) (\frac{n}{2})

and

k \geq 3

, then for sufficiently large

n

, the number of rainbow cliques

K_{k}

G

Ω (n^{k})

We also characterize the extremal graphs

G

without a rainbow clique

K_{k}

, for

k = 4, 5

, when

e (G) + c (G)

is maximum.

Our results not only address existing questions but also complete the findings of Ehard and Mohr (2020).

在边色图 G 中，彩虹簇 Kk 是 k 个顶点上的一个完整子图，其中所有的边都有不同的颜色。设 e(G) 和 c(G) 分别为 G 中的边数和颜色数。本文将证明，对于任意ɛ>0，如果 e(G)+c(G)≥(1+k-3k-2+2ɛ)n2 且 k≥3 ，那么对于足够大的 n，G 中彩虹小群 Kk 的数目为 Ω(nk)。我们还描述了在 k=4,5 时，e(G)+c(G) 最大时没有彩虹簇 Kk 的极值图 G 的特征。我们的结果不仅解决了现有问题，还完善了 Ehard 和 Mohr (2020) 的发现。

引用次数: 0

When (signless) Laplacian coefficients meet matchings of subdivision 当（无符号）拉普拉斯系数与细分匹配时

IF 1 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2024-11-13 DOI: 10.1016/j.ejc.2024.104087

Zhibin Du

Let

G

be a graph, whose subdivision is denoted by

S (G)

. Let

ϕ_{L} (G, x)

be the characteristic polynomial of the Laplacian matrix of

G

. In 1974, Kelmans and Chelnokov (1974) gave a graph theoretical interpretation for the coefficients of

ϕ_{L} (G, x)

, in terms of the spanning forests of

G

. In this paper, we present another graph theoretical interpretation of the Laplacian coefficients by using the matching numbers of

S (G)

, generalizing the cases of trees and unicyclic graphs, which were established by Zhou and Gutman (2008) and Chen and Yan (2021), respectively. Analogously, a graph theoretical interpretation of the signless Laplacian coefficients is also presented, whose previous graph theoretical interpretation is based on the so-called TU-subgraphs (the spanning subgraphs whose components are trees or odd-unicyclic graphs) due to Cvetković et al. (2007). Some formulas related to the number of spanning trees are also given.

设 G 是一个图，其细分图用 S(G) 表示。1974 年，Kelmans 和 Chelnokov（1974 年）用 G 的生成林给出了 ϕL(G,x) 系数的图论解释。在本文中，我们利用 S(G) 的匹配数提出了拉普拉奇系数的另一种图论解释，并推广了周和古特曼（2008 年）以及陈和严（2021 年）分别建立的树图和单环图的情况。与此类似，我们还提出了无符号拉普拉奇系数的图论解释，其先前的图论解释是基于 Cvetković 等人（2007 年）提出的所谓 TU 子图（其成分为树形或奇单环图的跨度子图）。此外，还给出了一些与生成树数量相关的公式。

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引用次数: 0

Freehedra are short 自由面很短

IF 1 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2024-11-12 DOI: 10.1016/j.ejc.2024.104084

Daria Poliakova

We prove the combinatorial property of shortness for freehedra. Note that associahedra, a related family of polytopes, are not short.

我们证明了自由曲面的组合短性。需要注意的是，相关联的多面体家族并不简短。

引用次数: 0

On the Erdős–Tuza–Valtr conjecture 关于厄尔多斯-图扎-瓦尔特猜想

IF 1 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2024-11-08 DOI: 10.1016/j.ejc.2024.104085

Jineon Baek

The Erdős–Szekeres conjecture states that any set of more than

2^{n - 2}

points in the plane with no three on a line contains the vertices of a convex

n

-gon. Erdős, Tuza, and Valtr strengthened the conjecture by stating that any set of more than

\sum_{i = n - b}^{a - 2} (\frac{n - 2}{i})

points in a plane either contains the vertices of a convex

n

-gon,

a

points lying on a concave downward curve, or

b

points lying on a concave upward curve. They also showed that the generalization is actually equivalent to the Erdős–Szekeres conjecture. We prove the first new case of the Erdős–Tuza–Valtr conjecture since the original 1935 paper of Erdős and Szekeres. Namely, we show that any set of

(\frac{n - 1}{2}) + 2

points in the plane with no three points on a line and no two points sharing the same

x

-coordinate either contains 4 points lying on a concave downward curve or the vertices of a convex

n

-gon. The proof is also formalized in Lean 4, a computer proof assistance, to ensure the correctness of the proof.

厄尔多斯-塞克雷斯猜想指出，平面中任何超过 2n-2 个点的集合，只要没有三点在一条直线上，就包含一个凸 n 形的顶点。Erdős、Tuza 和 Valtr 强化了这一猜想，指出平面上任何超过 ∑i=n-ba-2n-2i 个点的集合要么包含凸 n 形的顶点，要么包含位于向下凹曲线上的 a 个点，要么包含位于向上凹曲线上的 b 个点。他们还证明了这一推广实际上等同于厄尔多斯-塞克斯猜想。我们证明了 Erdős-Tuza-Valtr 猜想自 Erdős 和 Szekeres 于 1935 年发表论文以来的第一个新案例。也就是说，我们证明了平面上任何 n-12+2 个点的集合，其中没有三个点在一条直线上，也没有两个点共享相同的 x 坐标，要么包含位于向下凹曲线上的 4 个点，要么包含凸 n 形的顶点。为了确保证明的正确性，还用 Lean 4 这一计算机证明辅助工具将证明形式化。

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引用次数: 0

A combinatorial PROP for bialgebras 双桥的组合 PROP

IF 1 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2024-11-05 DOI: 10.1016/j.ejc.2024.104086

Jorge Becerra

It is a classical result that the category of finitely-generated free monoids serves as a PROP for commutative bialgebras. Attaching permutations to fix the order of multiplication, we construct an extension of this category that is equivalent to the PROP for bialgebras.

有限生成的自由单元范畴是交换双桥的 PROP，这是一个经典结果。通过附加排列来固定相乘的顺序，我们构建了这个范畴的一个扩展，它等同于双桥的 PROP。

引用次数: 0

Signed Mahonian polynomials on derangements in classical Weyl groups 经典韦尔群出射上的有符号马洪多项式

IF 1 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2024-11-04 DOI: 10.1016/j.ejc.2024.104083

Kathy Q. Ji , Dax T.X. Zhang

The polynomial of the major index

{maj}_{W} (σ)

over the subset

T

of the Coxeter group

W

is called the Mahonian polynomial over

T

, where

{maj}_{W} (σ)

is a Mahonian statistic of an element

σ \in T

, whereas the polynomial of the major index

{maj}_{W} (σ)

with the sign

{(- 1)}^{ℓ_{W} (σ)}

over the subset

T

is referred to as the signed Mahonian polynomial over

T

, where

ℓ_{W} (σ)

is the length of

σ \in T

. Gessel, Wachs, and Chow established formulas for the Mahonian polynomials over the sets of derangements in the symmetric group

S_{n}

and the hyperoctahedral group

B_{n}

. By extending Wachs’ approach and employing a refinement of Stanley’s shuffle theorem established in our recent paper (Ji and Zhang, 2024), we derive a formula for the Mahonian polynomials over the set of derangements in the even-signed permutation group

D_{n}

. This completes a picture which is now known for all the classical Weyl groups. Gessel–Simion, Adin–Gessel–Roichman, and Biagioli previously established formulas for the signed Mahonian polynomials over the classical Weyl groups. Building upon their formulas, we derive some new formulas for the signed Mahonian polynomials over the set of derangements in classical Weyl groups. As applications of the formulas for the (signed) Mahonian polynomials over the sets of derangements in the classical Weyl groups, we obtain enumerative formulas of the number of derangements in classical Weyl groups with even lengths.

考斯特群 W 的子集 T 上的主要指数 majW(σ) 的多项式称为 T 上的马洪多项式，其中 majW(σ) 是元素 σ∈T 的马洪统计量、而子集 T 上符号为 (-1)ℓW(σ) 的主要指数 majW(σ) 的多项式称为 T 上的带符号马洪多项式，其中 ℓW(σ) 是 σ∈T 的长度。Gessel、Wachs 和 Chow 建立了对称群 Sn 和超八面体群 Bn 中衍生集上的马洪多项式公式。通过扩展 Wachs 的方法，并利用我们最近的论文（Ji and Zhang, 2024）中建立的斯坦利洗牌定理的改进，我们推导出了偶符号置换群 Dn 的导数集上的马洪多项式公式。这完善了现在已知的所有经典韦尔群的情况。格塞尔-西米昂、阿丁-格塞尔-罗伊克曼和比亚乔利之前建立了经典韦尔群上有符号马洪多项式的公式。在他们的公式基础上，我们推导出了经典韦尔群中导数集上有符号马洪多项式的一些新公式。作为经典韦尔群导数集上（有符号）马洪多项式公式的应用，我们得到了经典韦尔群中偶数长度导数的枚举公式。

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引用次数: 0

Degree conditions for Ramsey goodness of paths 拉姆齐良好路径的程度条件

IF 1 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2024-10-18 DOI: 10.1016/j.ejc.2024.104082

Lucas Aragão , João Pedro Marciano , Walner Mendonça

A classical result of Chvátal implies that if

n \geq (r - 1) (t - 1) + 1

, then any colouring of the edges of

K_{n}

in red and blue contains either a monochromatic red

K_{r}

or a monochromatic blue

P_{t}

. We study a natural generalisation of his result, determining the exact minimum degree condition for a graph

G

n = (r - 1) (t - 1) + 1

vertices which guarantees that the same Ramsey property holds in

G

. In particular, using a slight generalisation of a result of Haxell, we show that

δ (G) \geq n - (t / 2)

suffices, and that this bound is best possible. We also use a classical result of Bollobás, Erdős, and Straus to prove a tight minimum degree condition in the case

r = 3

for all

n \geq 2 t - 1

Chvátal 的一个经典结果意味着，如果 n≥(r-1)(t-1)+1，那么 Kn 的任何红蓝边着色都包含一个单色红色 Kr 或一个单色蓝色 Pt。我们研究了他的结果的自然推广，确定了 n=(r-1)(t-1)+1 个顶点上的图 G 的精确最小度条件，该条件保证了相同的拉姆齐性质在 G 中成立。特别是，利用哈克赛尔结果的轻微推广，我们证明δ(G)≥n-t/2 就足够了，而且这个约束是最好的。我们还利用 Bollobás、Erdős 和 Straus 的经典结果，证明了在 r=3 的情况下，所有 n≥2t-1 的最小度条件。

引用次数: 0

On the faces of unigraphic 3-polytopes 关于单图式 3 多面体的面

IF 1 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2024-10-16 DOI: 10.1016/j.ejc.2024.104081

Riccardo W. Maffucci

A 3-polytope is a 3-connected, planar graph. It is called unigraphic if it does not share its vertex degree sequence with any other 3-polytope, up to graph isomorphism. The classification of unigraphic 3-polytopes appears to be a difficult problem.

In this paper we prove that, apart from pyramids, all unigraphic 3-polytopes have no

n

-gonal faces for

n \geq 10

. Our method involves defining several planar graph transformations on a given 3-polytope containing an

n

-gonal face with

n \geq 10

. The delicate part is to prove that, for every such 3-polytope, at least one of these transformations both preserves 3-connectivity, and is not an isomorphism.

3 多面体是一个 3 连接的平面图形。如果它的顶点度序列不与任何其他 3 多面体共享，直到图同构，那么它就被称为单图形。在本文中，我们证明了除金字塔外，所有单图形三多面体在 n≥10 时都没有 n 个球面。我们的方法是在一个给定的 3 多面体上定义几个平面图形变换，其中包含一个 n≥10 的 n 角面。最复杂的部分是证明，对于每一个这样的 3 多面体，这些变换中至少有一个既保留了 3 连通性，又不是同构。

引用次数: 0

Bounded unique representation bases for the integers 整数的有界唯一表示基

IF 1 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2024-10-16 DOI: 10.1016/j.ejc.2024.104080

Yong-Gao Chen, Jin-Hui Fang

<div><div>For a nonempty set <math><mi>A</mi></math> of integers and an integer <math><mi>n</mi></math>, let <math><mrow><msub><mrow><mi>r</mi></mrow><mrow><mi>A</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math> be the number of representations of <math><mrow><mi>n</mi><mo>=</mo><mi>a</mi><mo>+</mo><msup><mrow><mi>a</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow></math> with <math><mrow><mi>a</mi><mo>≤</mo><msup><mrow><mi>a</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow></math> and <math><mrow><mi>a</mi><mo>,</mo><msup><mrow><mi>a</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>∈</mo><mi>A</mi></mrow></math>, and let <math><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi>A</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math> be the number of representations of <math><mrow><mi>n</mi><mo>=</mo><mi>a</mi><mo>−</mo><msup><mrow><mi>a</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow></math> with <math><mrow><mi>a</mi><mo>,</mo><msup><mrow><mi>a</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>∈</mo><mi>A</mi></mrow></math>. Erdős and Turán (1941) posed the profound conjecture: if <math><mi>A</mi></math> is a set of positive integers such that <math><mrow><msub><mrow><mi>r</mi></mrow><mrow><mi>A</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>≥</mo><mn>1</mn></mrow></math> for all sufficiently large <math><mi>n</mi></math>, then <math><mrow><msub><mrow><mi>r</mi></mrow><mrow><mi>A</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math> is unbounded. Nešetřil and Serra (2004) introduced the notion of bounded sets and confirmed the Erdős–Turán conjecture for all bounded bases. Nathanson (2003) considered the existence of the set <math><mi>A</mi></math> with logarithmic growth such that <math><mrow><msub><mrow><mi>r</mi></mrow><mrow><mi>A</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>=</mo><mn>1</mn></mrow></math> for all integers <math><mi>n</mi></math>. In this paper, we prove that, for any positive function <math><mrow><mi>l</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></math> with <math><mrow><mi>l</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>→</mo><mn>0</mn></mrow></math> as <math><mrow><mi>x</mi><mo>→</mo><mi>∞</mi></mrow></math>, there is a bounded set <math><mi>A</mi></math> of integers such that <math><mrow><msub><mrow><mi>r</mi></mrow><mrow><mi>A</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>=</mo><mn>1</mn></mrow></math> for all integers <math><mi>n</mi></math> and <math><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi>A</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>=</mo><mn>1</mn></mrow></math> for all positi

对于非空整数集合 A 和整数 n，设 rA(n) 是 n=a+a′ 的表示数，其中 a≤a′ 和 a,a′∈A ；设 dA(n) 是 n=a-a′ 的表示数，其中 a,a′∈A 。厄尔多斯和图兰（1941）提出了一个深刻的猜想：如果 A 是一个正整数集合，对于所有足够大的 n，rA(n)≥1，那么 rA(n) 是无界的。Nešetřil 和 Serra (2004) 引入了有界集的概念，并证实了 Erdős-Turán 对所有有界基的猜想。Nathanson (2003) 考虑了具有对数增长的集合 A 的存在性，即对于所有整数 n，rA(n)=1。在本文中，我们证明了对于任何正函数 l(x)，当 x→∞ 时，l(x)→0，存在一个有界的整数集合 A，使得对于所有整数 n，rA(n)=1；对于所有正整数 n，dA(n)=1；对于所有足够大的 x，A(-x,x)≥l(x)logx，其中 A(-x,x) 是具有 -x≤a≤x 的元素 a∈A 的个数。

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全部化学•材料生命科学医学物理工程技术环境•农林材料科学地球科学法学管理学化学环境科学与生态学计算机科学教育学经济学农林科学人文科学生物学数学物理与天体物理心理学综合性期刊其他工业工程理学历史学农学文学信息工程

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