Journal of Combinatorial Theory Series B最新文献

英文中文

The codegree Turán density of 3-uniform tight cycles 3均匀紧循环的余度Turán密度

IF 1.4 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-08-07 DOI: 10.1016/j.jctb.2025.07.007

Simón Piga, Nicolás Sanhueza-Matamala, Mathias Schacht

Given any ε>0 we prove that every sufficiently large n-vertex 3-graph H where every pair of vertices is contained in at least (1/3+ε)n edges contains a copy of C10, i.e. the tight cycle on 10 vertices. In fact we obtain the same conclusion for every cycle Cℓ with ℓ≥19.

给定任意ε>；0，我们证明了每一个足够大的n顶点3-图H，其中每一对顶点至少包含（1/3+ε）n条边，其中包含C10的一个副本，即10个顶点上的紧环。事实上，对于每一个循环，我们都得到了相同的结论。

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 : 2025-07-01 Epub 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

Improved bounds for zero-sum cycles in Zpd 改进了Zpd中零和循环的边界

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-07-01 Epub Date: 2025-03-21 DOI: 10.1016/j.jctb.2025.03.001

Micha Christoph, Charlotte Knierim, Anders Martinsson, Raphael Steiner

<div><div>For a finite abelian group <math><mo>(</mo><mi>Γ</mi><mo>,</mo><mo>+</mo><mo>)</mo></math>, let <math><mi>n</mi><mo>(</mo><mi>Γ</mi><mo>)</mo></math> denote the smallest positive integer n such that for each labeling of the arcs of the complete digraph of order n using elements from Γ, there exists a directed cycle such that the total sum of the arc-labels along the cycle equals 0. Alon and Krivelevich initiated the study of the parameter <math><mi>n</mi><mo>(</mo><mo>⋅</mo><mo>)</mo></math> on cyclic groups and proved that <math><mi>n</mi><mo>(</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>)</mo><mo>=</mo><mi>O</mi><mo>(</mo><mi>q</mi><mi>log</mi><mo>⁡</mo><mi>q</mi><mo>)</mo></math>. Several improvements and generalizations of this bound have since been obtained, and an optimal bound in terms of the group order of the form <math><mi>n</mi><mo>(</mo><mi>Γ</mi><mo>)</mo><mo>≤</mo><mo>|</mo><mi>Γ</mi><mo>|</mo><mo>+</mo><mn>1</mn></math> was recently announced by Campbell, Gollin, Hendrey and the last author. While this bound is tight when the group Γ is cyclic, in cases when Γ is far from being cyclic, significant improvements on the bound can be made. In this direction, studying the prototypical case when <math><mi>Γ</mi><mo>=</mo><msubsup><mrow><mi>Z</mi></mrow><mrow><mi>p</mi></mrow><mrow><mi>d</mi></mrow></msubsup></math> is a power of a cyclic group of prime order, Letzter and Morrison [Journal of Combinatorial Theory Series B, 2024] showed that <math><mi>n</mi><mo>(</mo><msubsup><mrow><mi>Z</mi></mrow><mrow><mi>p</mi></mrow><mrow><mi>d</mi></mrow></msubsup><mo>)</mo><mo>≤</mo><mi>O</mi><mo>(</mo><mi>p</mi><mi>d</mi><msup><mrow><mo>(</mo><mi>log</mi><mo>⁡</mo><mi>d</mi><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math> and that <math><mi>n</mi><mo>(</mo><msubsup><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>d</mi></mrow></msubsup><mo>)</mo><mo>≤</mo><mi>O</mi><mo>(</mo><mi>d</mi><mi>log</mi><mo>⁡</mo><mi>d</mi><mo>)</mo></math>. They then posed the problem of proving an (asymptotically optimal) upper bound of <math><mi>n</mi><mo>(</mo><msubsup><mrow><mi>Z</mi></mrow><mrow><mi>p</mi></mrow><mrow><mi>d</mi></mrow></msubsup><mo>)</mo><mo>≤</mo><mi>O</mi><mo>(</mo><mi>p</mi><mi>d</mi><mo>)</mo></math> for all primes p and <math><mi>d</mi><mo>∈</mo><mi>N</mi></math>. In this paper, we solve this problem for <math><mi>p</mi><mo>=</mo><mn>2</mn></math> and improve their bound for all primes <math><mi>p</mi><mo>≥</mo><mn>3</mn></math> by proving <math><mi>n</mi><mo>(</mo><msubsup><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>d</mi></mrow></msubsup><mo>)</mo><mo>≤</mo><mn>5</mn><mi>d</mi></math> and <math><mi>n</mi><mo>(</mo><msubsup><mrow><mi>Z</mi></mrow><mrow><mi>p</mi></mrow>

对于有限阿贝尔群（Γ，+），设n（Γ）表示最小的正整数n，使得对于使用Γ中的元素标记n阶的完全有向图的每个弧，存在一个有向循环，使得沿循环的弧标记的总和等于0。Alon和Krivelevich开创了循环群上n（⋅）参数的研究，证明了n(Zq)=O（qlog）。此后，对这一界进行了若干改进和推广，最近由Campbell， Gollin， hendry和最后一位作者提出了n（Γ）≤|Γ|+1的群阶最优界。当组Γ是循环的时候，这个边界是紧的，而当Γ远不是循环的时候，可以对边界进行重大改进。在这个方向上，Letzter和Morrison [Journal of Combinatorial Theory Series B， 2024]研究了Γ=Zpd是一个素阶循环群幂的典型情况，证明了n(Zpd)≤O(pd(log d)2), n(Z2d)≤O（log d）。然后，他们提出了证明对于所有素数p和d∈n， n(Zpd)≤O（pd）的（渐近最优）上界的问题。本文通过证明n(Z2d)≤5d和n(Zpd)≤O(pdlog (d))，解决了p=2时的这一问题，并改进了p≥3时所有素数的界。当第一个边界决定n（Z2d）到5的乘法误差时，第二个边界紧到一个log (d)因子。此外，我们的结果表明，对于任意p和d， n(Zpd)=Θ（pd）的紧界是由著名的Jaeger， Linial， Payan和Tarsi在Zpd上的加性基猜想的一个（强形式）推导出来的。在证明这些结果的过程中，我们建立了Haxell在矩阵环境下的超图匹配结果的推广。具体地说，我们得到了超图中超边由一个拟阵的元素标记的匹配存在的充分条件，并得到了该超图中匹配中的边可以引出该拟阵的一组基。我们认为，这些声明具有独立的利益。

引用次数: 0

A splitter theorem on 3-connected binary matroids and inner fans 3连通二元拟阵和内扇的一个分岔定理

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-07-01 Epub Date: 2025-04-01 DOI: 10.1016/j.jctb.2025.03.004

João Paulo Costalonga

We establish a splitter type theorem for 3-connected binary matroids regarding elements whose contraction preserves a fixed 3-connected minor and the vertical 3-connectivity. We established that, for 3-connected simple binary matroids

N < M

, there is a disjoint family

{X_{1}, \dots, X_{n}} \subseteq 2^{E (M)}

such that

r (X_{1}) + \dots + r (X_{n}) = r (X_{1} \cup \dots \cup X_{n}) \geq r (M) - r (N)

, each

si (M / X_{i})

is 3-connected with an N-minor, and either

| X_{i} | = 1

or X is a special type of fan. We also establish a stronger version of this result under specific hypotheses. These results have several consequences, including the generalizations for binary matroids of some results about contractible edges in 3-connected graphs and some other structural results for graphs and binary matroids.

我们建立了关于元素的3连通二元拟阵的分裂型定理，这些元素的收缩保留了固定的3连通次元和垂直的3连通。我们建立了对于3连通的简单二元拟阵N<；M，存在一个不相交的族{X1，…，Xn}，使得r(X1)+⋯+r(Xn)=r(X1∪⋯∪Xn)≥r(M) - r(N)，每个si（M/Xi）与一个N次元3连通，且|Xi|=1或X是一个特殊类型的扇。我们还在特定的假设下建立了一个更强的版本。这些结果有几个结论，包括对3连通图中可缩边的一些结果在二元拟阵上的推广，以及图和二元拟阵的其他一些结构结果。

引用次数: 0

Sparse induced subgraphs of large treewidth 大树宽的稀疏诱导子图

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-07-01 Epub Date: 2025-03-20 DOI: 10.1016/j.jctb.2025.03.002

Édouard Bonnet

Motivated by an induced counterpart of treewidth sparsifiers (i.e., sparse subgraphs keeping the treewidth large) provided by the celebrated Grid Minor theorem of Robertson and Seymour (1986) [22] or by a classic result of Chekuri and Chuzhoy (2015) [5], we show that for any natural numbers t and w, and real

ε > 0

, there is an integer

W : = W (t, w, ε)

such that every graph with treewidth at least W and no

K_{t, t}

subgraph admits a 2-connected n-vertex induced subgraph with treewidth at least w and at most

(1 + ε) n

edges. The induced subgraph is either a subdivided wall, or its line graph, or a spanning supergraph of a subdivided biclique. This in particular extends a result of Weißauer (2019) [25] that graphs of large treewidth have a large biclique subgraph or a long induced cycle.

受著名的Robertson和Seymour（1986）的网格小定理（Grid Minor theorem）提供的树宽稀疏子图（即保持树宽较大的稀疏子图）的诱导对偶，或Chekuri和chuchoy（2015）的经典结果（[5]）的激励，我们表明，对于任何自然数t和w，以及实数ε>；0，存在一个整数W:=W(t， W,ε)，使得每个树宽至少W且没有Kt，t子图的图都存在一个树宽至少W且最多（1+ε）n条边的2连通n顶点诱导子图。诱导子图可以是细分壁面，也可以是细分壁面的线形图，也可以是细分壁面的生成超图。这特别扩展了Weißauer(2019)[25]的结果，即大树宽的图有一个大的双曲线子图或一个长诱导周期。

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引用次数: 0

Dynamics of cycles in polyhedra I: The isolation lemma 多面体循环动力学I：分离引理

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-07-01 Epub Date: 2024-05-16 DOI: 10.1016/j.jctb.2024.03.008

Jan Kessler , Jens M. Schmidt

<div><div>A cycle C of a graph G is isolating if every component of <math><mi>G</mi><mo>−</mo><mi>V</mi><mo>(</mo><mi>C</mi><mo>)</mo></math> consists of a single vertex. We show that isolating cycles in polyhedral graphs can be extended to larger ones: every isolating cycle C of length <math><mn>6</mn><mo>≤</mo><mo>|</mo><mi>E</mi><mo>(</mo><mi>C</mi><mo>)</mo><mo>|</mo><mo><</mo><mrow><mo>⌊</mo><mfrac><mrow><mn>2</mn></mrow><mrow><mn>3</mn></mrow></mfrac><mo>(</mo><mo>|</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>|</mo><mo>+</mo><mn>4</mn><mo>)</mo><mo>⌋</mo></mrow></math> implies an isolating cycle <math><msup><mrow><mi>C</mi></mrow><mrow><mo>′</mo></mrow></msup></math> of larger length that contains <math><mi>V</mi><mo>(</mo><mi>C</mi><mo>)</mo></math>. By “hopping” iteratively to such larger cycles, we obtain a powerful and very general inductive motor for proving long cycles and computing them (we will give an algorithm with quadratic running time). This is the first step towards the so far elusive quest of finding a universal induction that captures longest cycles of polyhedral graph classes.</div><div>Our motor provides also a method to prove linear lower bounds on the length of Tutte cycles, as <math><msup><mrow><mi>C</mi></mrow><mrow><mo>′</mo></mrow></msup></math> will be a Tutte cycle of G if C is. We prove in addition that <math><mo>|</mo><mi>E</mi><mo>(</mo><msup><mrow><mi>C</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>)</mo><mo>|</mo><mo>≤</mo><mo>|</mo><mi>E</mi><mo>(</mo><mi>C</mi><mo>)</mo><mo>|</mo><mo>+</mo><mn>3</mn></math> if G contains no face of size five, which gives a new tool for results about cycle spectra, and provides evidence that faces of size five may obstruct many different cycle lengths. As a sample application, we test our motor on the following so far unsettled conjecture about essentially 4-connected graphs.</div><div>A planar graph is essentially 4-connected if it is 3-connected and every of its 3-separators is the neighborhood of a single vertex. Essentially 4-connected graphs have been thoroughly investigated throughout literature as the subject of Hamiltonicity studies. Jackson and Wormald proved that every essentially 4-connected planar graph G on n vertices contains a cycle of length at least <math><mfrac><mrow><mn>2</mn></mrow><mrow><mn>5</mn></mrow></mfrac><mo>(</mo><mi>n</mi><mo>+</mo><mn>2</mn><mo>)</mo></math>, and this result has recently been improved multiple times, culminating in the lower bound <math><mfrac><mrow><mn>5</mn></mrow><mrow><mn>8</mn></mrow></mfrac><mo>(</mo><mi>n</mi><mo>+</mo><mn>2</mn><mo>)</mo></math>. However, the currently best known upper bound is given by an infinite family of such graphs in which no graph G contains a cycle that is longer than <

如果图G的每个分量G−V(C)都由单个顶点组成，则图G的循环C是孤立的。我们证明了多面体图中的隔离环可以推广为更大的隔离环：每一个长度为6≤|E(C)|<；⌊23（|V(G)|+4）⌋的隔离环C ‘都隐含一个包含V(C)的更大长度的隔离环C ’。通过迭代地“跳跃”到如此大的周期，我们得到了一个强大而非常通用的感应电机，用于证明长周期并计算它们（我们将给出一个运行时间为二次的算法）。这是迄今为止难以捉摸的寻找捕获多面体图类最长周期的普遍归纳的第一步。我们的电机还提供了一种方法来证明Tutte周期长度的线性下界，因为C '将是G的Tutte周期，如果C为。此外，我们还证明了如果G不包含5号面，则|E(C ')|≤|E(C)|+3，这为循环光谱的结果提供了新的工具，并为5号面可能阻碍许多不同的循环长度提供了证据。作为一个示例应用程序，我们在以下关于本质上是4连通图的尚未解决的猜想上测试我们的电机。如果一个平面图形是3连通的，并且它的每个3分隔符都是单个顶点的邻域，那么它本质上是4连通的。基本上，在整个文献中，4连通图作为哈密顿性研究的主题已经被彻底地研究过。Jackson和Wormald证明了n个顶点上的每一个本质上是4连通的平面图G都包含一个长度至少为25（n+2）的循环，这个结果最近得到了多次改进，最终得到了下界58（n+2）。然而，目前已知的上界是由这样的图的无限族给出的，其中没有图G包含大于⌊23（n+4）⌋的循环；这个上界仍然是不匹配的。利用隔离循环，我们改进了下界以匹配上界。这解决了一个长期悬而未决的问题，即确定本质上是4连通的平面图的周长。我们所有的结果都很紧凑。

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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 : 2025-07-01 Epub 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

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 : 2025-07-01 Epub 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

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 : 2025-07-01 Epub 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

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 : 2025-07-01 Epub 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

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