Journal of Combinatorial Theory Series B最新文献

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Some results and problems on tournament structure 关于赛事结构的一些结果和问题

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-02-28 DOI: 10.1016/j.jctb.2025.02.002

Tung Nguyen , Alex Scott , Paul Seymour

This paper is a survey of results and problems related to the following question: is it true that if G is a tournament with sufficiently large chromatic number, then G has two vertex-disjoint subtournaments

A, B

, both with large chromatic number, such that all edges between them are directed from A to B? We describe what we know about this question, and report some progress on several other related questions, on tournament colouring and domination.

本文研究了以下问题的结果和问题：如果G是一个具有足够大色数的锦标赛，那么G是否有两个顶点不相交的子锦标赛a，B，它们都具有较大的色数，并且它们之间的所有边都从a指向B？我们描述了我们对这个问题的了解，并报告了一些其他相关问题的进展，关于比赛的颜色和统治。

引用次数: 0

Ramsey numbers of bounded degree trees versus general graphs 有界度树与一般图的Ramsey数

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-02-21 DOI: 10.1016/j.jctb.2025.02.004

Richard Montgomery , Matías Pavez-Signé , Jun Yan

For every

k \geq 2

and Δ, we prove that there exists a constant

C_{Δ, k}

such that the following holds. For every graph H with

χ (H) = k

and every tree T with at least

C_{Δ, k} | H |

vertices and maximum degree at most Δ, the Ramsey number

R (T, H)

(k - 1) (| T | - 1) + σ (H)

, where

σ (H)

is the size of a smallest colour class across all proper k-colourings of H. This is tight up to the value of

C_{Δ, k}

, and confirms a conjecture of Balla, Pokrovskiy, and Sudakov.

对于每一个k≥2和Δ，我们证明存在一个常数CΔ，k，使得下式成立。对于每一个χ(H)=k的图H和每一个至少有CΔ，k|H|顶点且最大度不超过Δ的树T，拉姆齐数R（T，H）等于（k−1）(|T|−1)+σ(H)，其中σ(H)是横跨H的所有适当的k-着色的最小颜色类的大小。这紧达CΔ，k的值，并证实了Balla， Pokrovskiy和Sudakov的一个猜想。

引用次数: 0

Tree amalgamations and quasi-isometries 树合并和准等距

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-02-14 DOI: 10.1016/j.jctb.2025.02.003

Matthias Hamann

We investigate the connections between tree amalgamations and quasi-isometries. In particular, we prove that the quasi-isometry type of multi-ended accessible quasi-transitive connected locally finite graphs is determined by the quasi-isometry type of their one-ended factors in any of their terminal factorisations. Our results carry over theorems of Papasoglu and Whyte on quasi-isometries between multi-ended groups to those between multi-ended graphs. In the end, we discuss the impact of our results to a question of Woess.

我们研究了树合并和拟等距之间的联系。特别地，我们证明了多端可达拟传递连通局部有限图的拟等距型是由其任意终端分解中的单端因子的拟等距型决定的。我们的结果将Papasoglu和Whyte关于多端群间拟等距的定理推广到多端图间的拟等距定理。最后，我们讨论了我们的结果对一个Woess问题的影响。

引用次数: 0

A counterexample to the coarse Menger conjecture 粗糙门格尔猜想的反例

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-02-13 DOI: 10.1016/j.jctb.2025.01.004

Tung Nguyen , Alex Scott , Paul Seymour

Menger's well-known theorem from 1927 characterizes when it is possible to find k vertex-disjoint paths between two sets of vertices in a graph G. Recently, Georgakopoulos and Papasoglu and, independently, Albrechtsen, Huynh, Jacobs, Knappe and Wollan conjectured a coarse analogue of Menger's theorem, when the k paths are required to be pairwise at some distance at least d. The result is known for

k \leq 2

, but we will show that it is false for all

k \geq 3

, even if G is constrained to have maximum degree at most three. We also give a simpler proof of the result when

k = 2

最近，Georgakopoulos和Papasoglu，以及独立的Albrechtsen， Huynh, Jacobs， Knappe和Wollan推测了门格尔定理的一个粗略的类似，当k路径被要求在至少d的距离上配对时，结果是已知的k≤2，但我们将证明它对所有k≥3都是错误的。即使G被约束最大度不超过3。我们还给出了k=2时的一个更简单的证明。

引用次数: 0

Clustered coloring of (path + 2K1)-free graphs on surfaces 曲面上（路径 + 2K1）自由图的聚类着色

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-02-13 DOI: 10.1016/j.jctb.2025.02.001

Zdeněk Dvořák

Esperet and Joret proved that planar graphs with bounded maximum degree are 3-colorable with bounded clustering. Liu and Wood asked whether the conclusion holds with the assumption of the bounded maximum degree replaced by assuming that no two vertices have many common neighbors. We answer this question in positive, in the following stronger form: Let

P_{t}^{″}

be the complete join of two isolated vertices with a path on t vertices. For any surface Σ, a subgraph-closed class of graphs drawn on Σ is 3-choosable with bounded clustering if and only if there exists t such that

P_{t}^{″}

does not belong to the class.

Esperet和Joret证明了具有有界最大度的平面图具有有界聚类的3色性。Liu和Wood提出了一个问题，当最大度有界的假设被没有两个顶点有很多共同邻居的假设所取代时，结论是否成立。我们正回答这个问题，用以下更强的形式：设Pt″是两个孤立顶点的完全连接，在t个顶点上有一条路径。对于任意曲面Σ，当且仅当存在t使得Pt″不属于该类时，在Σ上绘制的图的子图闭类具有有界聚类是可3选的。

引用次数: 0

Ascending subgraph decomposition 升子图分解

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-02-12 DOI: 10.1016/j.jctb.2025.01.003

Kyriakos Katsamaktsis , Shoham Letzter , Alexey Pokrovskiy , Benny Sudakov

A typical theme for many well-known decomposition problems is to show that some obvious necessary conditions for decomposing a graph G into copies of

H_{1}, \dots, H_{m}

are also sufficient. One such problem was posed in 1987, by Alavi, Boals, Chartrand, Erdős, and Oellerman. They conjectured that the edges of every graph with

(\begin{matrix} m + 1 \\ 2 \end{matrix})

edges can be decomposed into subgraphs

H_{1}, \dots, H_{m}

such that each

H_{i}

has i edges and is isomorphic to a subgraph of

H_{i + 1}

. In this paper we prove this conjecture for sufficiently large m.

对于许多著名的分解问题，一个典型的主题是证明将图G分解成H1，…，Hm的副本的一些明显的必要条件也是充分的。1987年，Alavi、Boals、Chartrand、Erdős和Oellerman提出了一个这样的问题。他们推测，每个有（m+12）条边的图的边都可以分解成子图H1，…，Hm，使得每个Hi都有i条边，并且与Hi+1的子图同构。本文在m足够大的情况下证明了这个猜想。

引用次数: 0

Every d(d + 1)-connected graph is globally rigid in Rd 每个d（d + 1）连通图在Rd中是全局刚性的

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-02-12 DOI: 10.1016/j.jctb.2025.01.005

Soma Villányi

Using a probabilistic method, we prove that

d (d + 1)

-connected graphs are rigid in

R^{d}

, a conjecture of Lovász and Yemini. Then, using recent results on weakly globally linked pairs, we modify our argument to prove that

d (d + 1)

-connected graphs are globally rigid, too, a conjecture of Connelly, Jordán and Whiteley. The constant

d (d + 1)

is best possible.

用概率方法证明了d（d+1）连通图在Rd上是刚性的，Rd是Lovász和Yemini的一个猜想。然后，利用弱全局连接对上的最新结果，我们修正了我们的论证，证明d（d+1）连通图也是全局刚性的，这是Connelly， Jordán和Whiteley的一个猜想。常数d（d+1）是最好的。

引用次数: 0

Cumulant expansion for counting Eulerian orientations 欧拉取向计数的累积展开

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-01-24 DOI: 10.1016/j.jctb.2025.01.002

Mikhail Isaev , Brendan D. McKay , Rui-Ray Zhang

An Eulerian orientation is an orientation of the edges of a graph such that every vertex is balanced: its in-degree equals its out-degree. Counting Eulerian orientations corresponds to the crucial partition function in so-called “ice-type models” in statistical physics and is known to be hard for general graphs. For all graphs with good expansion properties and degrees larger than

\log^{8} n

, we derive an asymptotic expansion for this count that approximates it to precision

O (n^{- c})

for arbitrarily large c, where n is the number of vertices. The proof relies on a new tail bound for the cumulant expansion of the Laplace transform, which is of independent interest.

欧拉方向是一个图的边的方向，使得每个顶点都是平衡的：它的入度等于它的出度。计算欧拉方向对应于统计物理中所谓的“冰型模型”中的关键配分函数，并且对于一般图形来说是困难的。对于所有具有良好展开性且度大于log8 n的图，我们推导出该计数的渐近展开式，对于任意大的c，该计数近似于精度O(n - c)，其中n是顶点数。证明依赖于拉普拉斯变换的累积展开的一个新的尾界，这是一个独立的兴趣。

引用次数: 0

Toward a density Corrádi–Hajnal theorem for degenerate hypergraphs 关于退化超图的密度Corrádi-Hajnal定理

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-01-23 DOI: 10.1016/j.jctb.2025.01.001

Jianfeng Hou , Caiyun Hu , Heng Li , Xizhi Liu , Caihong Yang , Yixiao Zhang

Given an r-graph F with

r \geq 2

, let

ex (n, (t + 1) F)

denote the maximum number of edges in an n-vertex r-graph with at most t pairwise vertex-disjoint copies of F. Extending several old results and complementing prior work [34] on nondegenerate hypergraphs, we initiate a systematic study on

ex (n, (t + 1) F)

for degenerate hypergraphs F.

For a broad class of degenerate hypergraphs F, we present near-optimal upper bounds for

ex (n, (t + 1) F)

when n is sufficiently large and t lies in intervals

[0, \frac{ε \cdot ex (n, F)}{n^{r - 1}}]

[\frac{ex (n, F)}{ε n^{r - 1}}, ε n]

, and

[(1 - ε) \frac{n}{v (F)}, \frac{n}{v (F)}]

, where

ε > 0

is a constant depending only on F. Our results reveal very different structures for extremal constructions across the three intervals, and we provide characterizations of extremal constructions within the first interval. Additionally, we characterize extremal constructions within the second interval for graphs. Our proof for the first interval also applies to a special class of nondegenerate hypergraphs, including those with undetermined Turán densities, partially improving a result in [34].

给定一个r≥2的r-图F，设ex（n,(t+1)F）表示一个n顶点的r-图的最大边数，该r-图最多有t个对顶点不相交的副本F。我们扩展了几个旧的结果，并补充了先前关于非退化超图的工作[34]，系统地研究了退化超图F的ex（n,(t+1)F）。对于一类广义的退化超图F，我们给出了当n足够大且t位于区间[0，ε·ex(n，F)nr - 1]时ex（n,(t+1)F）的近最优上界。[ex(n，F)εnr−1，εn]和[(1 - ε)nv(F),nv(F)]，其中ε>；0是仅依赖于F的常数。我们的结果揭示了三个区间中极值结构的非常不同的结构，并在第一个区间内给出了极值结构的表征。此外，我们描述了图在第二区间内的极值结构。我们对第一个区间的证明也适用于一类特殊的非退化超图，包括那些具有待定Turán密度的超图，部分地改进了[34]的结果。

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引用次数: 0

The next case of Andrásfai's conjecture Andrásfai猜想的下一个例子

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-01-14 DOI: 10.1016/j.jctb.2024.12.010

Tomasz Łuczak , Joanna Polcyn , Christian Reiher

Let

ex (n, s)

denote the maximum number of edges in a triangle-free graph on n vertices which contains no independent sets larger than s. The behaviour of

ex (n, s)

was first studied by Andrásfai, who conjectured that for

s > n / 3

this function is determined by appropriately chosen blow-ups of so called Andrásfai graphs. Moreover, he proved

ex (n, s) = n^{2} - 4 n s + 5 s^{2}

for

s / n \in [2 / 5, 1 / 2]

and in earlier work we obtained

ex (n, s) = 3 n^{2} - 15 n s + 20 s^{2}

for

s / n \in [3 / 8, 2 / 5]

. Here we make the next step in the quest to settle Andrásfai's conjecture by proving

ex (n, s) = 6 n^{2} - 32 n s + 44 s^{2}

for

s / n \in [4 / 11, 3 / 8]

设ex（n,s）表示无三角形图在n个顶点上的最大边数，其中不包含大于s的独立集。Andrásfai首先研究了ex（n,s）的行为，他推测对于s>；n/3，该函数由适当选择的所谓Andrásfai图的放大决定。此外，他证明了对于s/n∈[2/5,1/2],ex(n,s)=n2−4ns+5s2，在之前的工作中，我们得到了对于s/n∈[3/8,2/5],ex(n,s)=3n2−15ns+20s2。在这里，我们通过证明s/n∈[4/11,3/8]的ex(n,s)=6n2−32ns+44s2来解决Andrásfai猜想的下一步。

引用次数: 0

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