Journal of Combinatorial Theory Series B最新文献

英文中文

Cycle matroids of graphings: From convergence to duality 图的循环拟阵：从收敛到对偶

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-12-15 DOI: 10.1016/j.jctb.2025.12.003

Kristóf Bérczi , Márton Borbényi , László Lovász , László Márton Tóth

A recent line of research has concentrated on exploring the links between analytic and combinatorial theories of submodularity, uncovering several key connections between them. In this context, Lovász initiated the study of matroids from an analytic point of view and introduced the cycle matroid of a graphing [23]. Motivated by the limit theory of graphs, the authors introduced a form of right-convergence, called quotient-convergence, for a sequence of submodular setfunctions, leading to a notion of convergence for matroids through their rank functions [5].

In this paper, we study the connection between local-global convergence of graphs and quotient-convergence of their cycle matroids. We characterize the exposed points of associated convex sets, forming an analytic counterpart of matroid independence- and base-polytopes. Finally, we consider dual planar graphings and show that the cycle matroid of one is the cocycle matroid of its dual if and only if the underlying graphings are hyperfinite.

最近的一项研究集中在探索亚模块化的分析理论和组合理论之间的联系，揭示了它们之间的几个关键联系。在此背景下，Lovász从解析的角度开始了对拟阵的研究，并引入了一个绘图[23]的循环拟阵。在图的极限理论的启发下，作者引入了一组次模集合函数的右收敛形式，称为商收敛，从而引出了拟阵通过秩函数[5]收敛的概念。

引用次数: 0

Colorings of k-sets with low discrepancy on small sets 小集上低差异k集的着色

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-12-10 DOI: 10.1016/j.jctb.2025.12.001

Pavel Pudlák , Vojtěch Rödl

For

0 < δ \leq 1

, let

R_{k} (m; δ)

denote the smallest N such that every coloring of k-element subsets by two colors yields an m-element set M with relative discrepancy δ, which means that one color class has at least

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

elements. The number

R_{k} (m; δ)

may be viewed as an extension of the usual k-hypergraph Ramsey number because

R_{k} (m) = R_{k} (m, 1)

. Our main result is the following theorem. For some constants

c, k_{0}

, and

ε > 0

, and for all

k \geq k_{0}

c \log k \leq n \leq k / 11

R_{k} (k + n; 2^{- ε n}) \geq {tw}_{⌊ k / n ⌋} (2) .

In particular, for

n = ⌈ c \log k ⌉

, we get a tower of height

δ k / \log k

and relative discrepancy polynomial in k.

对于0<；δ≤1，设Rk（m;δ）表示最小的N，使得k元素子集每用两种颜色着色得到一个相对差值为δ的m元素集合m，这意味着一个颜色类至少有（1+δ2）（mk）个元素。由于Rk(m)=Rk(m,1)，所以Rk（m;δ）可以看作是通常的k-超图拉姆齐数的扩展。我们的主要结果是下面的定理。对于某些常数c，k0, ε>0，且对于所有k≥k0， clog (k)≤n≤k/11,Rk(k+n;2 - εn)≥tw⌊k/n⌋(2)。特别地，对于n=∑clog²k，我们得到一个高度为δk/log²k的塔和k的相对差异多项式。

引用次数: 0

Diameter bounds for distance-regular graphs via long-scale Ollivier Ricci curvature 通过长尺度奥利维耶·里奇曲率的距离正则图的直径界

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-12-10 DOI: 10.1016/j.jctb.2025.12.002

Kaizhe Chen , Shiping Liu

In this paper, we derive new sharp diameter bounds for distance regular graphs, which better answer a problem raised by Neumaier and Penjić in many cases. Our proof is built upon a relation between the diameter and long-scale Ollivier Ricci curvature of a graph, which can be considered as an improvement of the discrete Bonnet-Myers theorem. Our method further leads to significant improvements of existing diameter bounds for amply regular graphs and

(s, c, a, k)

-graphs.

本文给出了距离正则图的新的尖锐直径界，较好地回答了Neumaier和penjiki在许多情况下提出的问题。我们的证明是建立在图的直径和长尺度奥利维耶·里奇曲率之间的关系上的，它可以被认为是对离散的邦纳-迈尔斯定理的改进。我们的方法进一步显著改进了现有的充分正则图和(s,c,a,k)-图的直径界。

引用次数: 0

On the size of two minimal linkages 两个最小连杆的大小

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-12-08 DOI: 10.1016/j.jctb.2025.11.007

Koyo Hayashi , Ken-ichi Kawarabayashi , Daiki Kobayashi

We prove that for any two positive integers

k_{1}

and

k_{2}

, if G is a graph,

S_{1}, T_{1}, S_{2}, T_{2}

are vertex-subsets of G, and G is edge-minimal with respect to the condition that for

i = 1, 2

there are

k_{i}

disjoint paths of G between

S_{i}

and

T_{i}

, then G contains at most

12 k_{1} k_{2}

vertices of degree at least three. This bound is optimal up to a constant factor, as a

k_{1} \times k_{2}

-grid G shows.

The degree-3 treewidth sparsifier theorem, proved by Chekuri and Chuzhoy (2015), states that for any graph G of treewidth at least k, there is a subcubic subgraph of G that has treewidth

Ω (k / poly \log k)

and contains

O (k^{4})

vertices of degree three. Our result, together with their proof techniques, reduces the bound

O (k^{4})

in this theorem to

O (k^{2})

, solving their conjecture.

我们证明了对于任意两个正整数k1和k2，如果G是一个图，S1,T1,S2，T2是G的顶点子集，并且G是边极小的，且对于i=1,2时，G在Si和Ti之间有ki条不相交的路径，则G最多包含12k1k2个至少3次的顶点。这个界是最优的，直到一个常数因子，如k1×k2-grid G所示。

引用次数: 0

Reduced bandwidth: A qualitative strengthening of twin-width in minor-closed classes (and beyond) 减少带宽：在小封闭类（及其他）中定性地加强双宽度

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-12-05 DOI: 10.1016/j.jctb.2025.11.010

Édouard Bonnet , O-joung Kwon , David R. Wood

In a reduction sequence of a graph, vertices are successively identified until the graph has one vertex. At each step, when identifying u and v, each edge incident to exactly one of u and v is coloured red. Bonnet, Kim, Thomassé and Watrigant (2022) [19] defined the twin-width of a graph G to be the minimum integer k such that there is a reduction sequence of G in which every red graph has maximum degree at most k. For any graph parameter f, we define the reduced f of a graph G to be the minimum integer k such that there is a reduction sequence of G in which every red graph has f at most k. Our focus is on graph classes with bounded reduced bandwidth, which implies and is stronger than bounded twin-width (reduced maximum degree). We show that every proper minor-closed class has bounded reduced bandwidth, which is qualitatively stronger than an analogous result of Bonnet et al. for bounded twin-width. In many instances, we also make quantitative improvements. For example, all previous upper bounds on the twin-width of planar graphs were at least 2¹⁰⁰⁰. We show that planar graphs have reduced bandwidth at most 466 and twin-width at most 583. Our bounds for graphs of Euler genus γ are

O (γ)

. Lastly, we show that fixed powers of graphs in a proper minor-closed class have bounded reduced bandwidth (irrespective of the degree of the vertices). In particular, we show that map graphs of Euler genus γ have reduced bandwidth

O (γ^{4})

. Lastly, we separate twin-width and reduced bandwidth by showing that any infinite class of expanders excluding a fixed complete bipartite subgraph has unbounded reduced bandwidth, while there are bounded-degree expanders with twin-width at most 6.

在图的约简序列中，连续地识别顶点，直到图只有一个顶点。在每一步中，当确定u和v时，与u和v中的一个相关的每条边都被涂成红色。阀盖,金正日,Thomasse Watrigant(2022)[19]定义了图G的twin-width最小整数k,这样有一个减少序列图G的每一个红色最大程度最多k。任何图参数f,我们定义一个图G的f的最小整数k,这样有一个减少序列图G的每一个红色f最多k。我们的重点是与有界图形类减少了带宽,这意味着并且比有界双宽度（减小最大度）更强。我们证明了每个适当的小闭类都有有界的减少带宽，这在定性上强于Bonnet等人关于有界双宽度的类似结果。在许多情况下，我们还进行了定量改进。例如，以前所有平面图双宽度的上界都至少为21000。我们表明，平面图形的带宽最多减少466，双宽度最多减少583。欧拉属γ图的界是O（γ）。最后，我们证明了在适当的小闭类中的固定幂图具有有限的减少带宽（与顶点的程度无关）。特别地，我们证明了欧拉属γ的映射图减少了带宽O（γ4）。最后，我们通过证明除固定完全二部子图外的任何无限类展板具有无界的缩减带宽，而存在双宽度最多为6的有界度展板来分离双宽度和缩减带宽。

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引用次数: 0

Cutting corners 偷工减料

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-12-05 DOI: 10.1016/j.jctb.2025.11.008

Andrey Kupavskii , Arsenii Sagdeev , Dmitrii Zakharov

<div><div>We say that a subset <math><mi>M</mi></math> of <math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math> is exponentially Ramsey if there exists <math><mi>ε</mi><mo>></mo><mn>0</mn></math> and <math><msub><mrow><mi>n</mi></mrow><mrow><mn>0</mn></mrow></msub></math> such that <math><mi>χ</mi><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>,</mo><mi>M</mi><mo>)</mo><mo>≥</mo><msup><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mi>ε</mi><mo>)</mo></mrow><mrow><mi>n</mi></mrow></msup></math> for any <math><mi>n</mi><mo>></mo><msub><mrow><mi>n</mi></mrow><mrow><mn>0</mn></mrow></msub></math>, where <math><mi>χ</mi><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>,</mo><mi>M</mi><mo>)</mo></math> stands for the minimum number of colors in a coloring of <math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math> such that no copy of <math><mi>M</mi></math> is monochromatic. One important result in Euclidean Ramsey theory is due to Frankl and Rödl, and states the following (under some mild extra conditions): if both <math><msub><mrow><mi>N</mi></mrow><mrow><mn>1</mn></mrow></msub></math> and <math><msub><mrow><mi>N</mi></mrow><mrow><mn>2</mn></mrow></msub></math> are exponentially Ramsey then so is their Cartesian product. Applied several times to simple two-point sets <math><msub><mrow><mi>N</mi></mrow><mrow><mi>i</mi></mrow></msub></math>, this result implies that any subset <math><mi>M</mi></math> of a ‘hyperrectangle’ <math><msub><mrow><mi>N</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>×</mo><mo>⋯</mo><mo>×</mo><msub><mrow><mi>N</mi></mrow><mrow><mi>k</mi></mrow></msub></math> is exponentially Ramsey.</div><div>However, generally, such ‘embeddings’ of <math><mi>M</mi></math> result in very inefficient bounds on the aforementioned ε. In this paper, we present another way of combining exponentially Ramsey sets, which gives much better estimates in some important cases. In particular, we show that the chromatic number of <math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math> with a forbidden equilateral triangle satisfies <math><mi>χ</mi><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>,</mo><mo>△</mo><mo>)</mo><mo>≥</mo><msup><mrow><mo>(</mo><mn>1.0742</mn><mo>.</mo><mo>.</mo><mo>.</mo><mo>+</mo><mi>o</mi><mo>(</mo><mn>1</mn><mo>)</mo><mo>)</mo></mrow><mrow><mi>n</mi></mrow></msup></math>, greatly improving upon the previous constant 1.0144. We also obtain similar strong results for regular simplices of larger dimensions, as well as for related geometric Ramsey-type questions in Manhattan norm.</div><div>We then show that the same technique implies several interesting corollaries in other combi

我们说Rn的一个子集M是指数拉姆齐的，如果存在ε>；0和n，使得χ(Rn，M)≥(1+ε)n对于任何n>；n，其中χ（Rn，M）表示在Rn的一个着色中使M没有一个副本是单色的最小颜色数。欧几里得拉姆齐理论中的一个重要结果是由Frankl和Rödl得出的，并且陈述如下（在一些温和的额外条件下）：如果N1和N2都是指数拉姆齐，那么它们的笛卡尔积也是指数拉姆齐。多次应用于简单两点集Ni，这一结果意味着“超矩形”n1x⋯×Nk的任何子集M都是指数拉姆齐。然而，一般来说，M的这种“嵌入”导致前面提到的ε的边界非常低效。在本文中，我们提出了另一种组合指数拉姆齐集的方法，它在一些重要的情况下给出了更好的估计。特别地，我们证明了带有禁止等边三角形的Rn的色数满足χ（Rn，△）≥(1.0742…+o(1))n，大大改善了之前的常数1.0144。对于较大维度的正则简式，以及在曼哈顿范数中相关的几何ramsey型问题，我们也得到了类似的强结果。然后，我们证明了同样的技术在其他组合问题中隐含了几个有趣的推论。特别地，我们给出了一个族F∧2[n]大小的显式上界，该族不包含弱k-向日葵，即不包含具有相同大小的成对相交的k个集合的集合。这个界改进了先前已知的所有k≥4的结果。最后，我们还从Frankl和Wilson的早期结果中提出了（另一个）著名的Frankl-Rödl定理的简单演绎。它给出了可能是已知最短的Frankl证明和Rödl最有效界的结果。

引用次数: 0

On digraphs without onion star immersions 在没有洋葱星浸没的有向图上

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-12-04 DOI: 10.1016/j.jctb.2025.11.009

Łukasz Bożyk , Oscar Defrain , Karolina Okrasa , Michał Pilipczuk

The t-onion star is the digraph obtained from a star with 2t leaves by replacing every edge by a triple of arcs, where in t triples we orient two arcs away from the center, and in the remaining t triples we orient two arcs towards the center. Note that the t-onion star contains, as an immersion, every digraph on t vertices where each vertex has outdegree at most 2 and indegree at most 1, or vice versa.

We investigate the structure in digraphs that exclude a fixed onion star as an immersion. The main discovery is that in such digraphs, for some duality statements true in the undirected setting we can prove their directed analogues. More specifically, we show the next two statements.

•
There is a function $f : N \to N$ satisfying the following: If a digraph D contains a set X of $2 t + 1$ vertices such that for any $x, y \in X$ there are $f (t)$ arc-disjoint paths from x to y, then D contains the t-onion star as an immersion.
•
There is a function $g : N \times N \to N$ satisfying the following: If x and y is a pair of vertices in a digraph D such that there are at least $g (t, k)$ arc-disjoint paths from x to y and there are at least $g (t, k)$ arc-disjoint paths from y to x, then either D contains the t-onion star as an immersion, or there is a family of 2k pairwise arc-disjoint paths with k paths from x to y and k paths from y to x.

t-洋葱星形是从一个有2t个叶子的星形中得到的有向图，通过将每条边替换为三组圆弧，其中在t个三组中，我们将两个圆弧定位在远离中心的位置，在剩下的t个三组中，我们将两个圆弧定位在靠近中心的位置。注意，t-洋葱星形包含t个顶点上的每个有向图，其中每个顶点最多有2度，最多有1度，反之亦然。我们研究了有向图中排除固定洋葱星作为浸没的结构。主要的发现是，在这样的有向图中，对于某些对偶命题在无向集合中为真，我们可以证明它们的有向类似。更具体地说，我们将展示下面两个语句。•存在一个函数f:N→N满足以下条件：如果有向图D包含2t+1个顶点的集合X，使得对于任意X，y∈X有f(t)条从X到y的弧不相交路径，则D包含t-洋葱星作为浸入式。•有一个函数g:N×N→N满足以下条件：如果x和y是有向图D中的一对顶点，使得从x到y至少有g（t,k）条弧不相交路径，并且从y到x至少有g（t,k）条弧不相交路径，那么D要么包含t-洋葱星作为浸没，要么存在2k对弧不相交路径族，其中x到y有k条路径，y到x有k条路径。

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引用次数: 0

Inducibility in H-free graphs and inducibility of Turán graphs 无h图的诱导性和Turán图的诱导性

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-12-04 DOI: 10.1016/j.jctb.2025.11.006

Raphael Yuster

<div><div>For graphs F and H, let <math><mi>i</mi><mo>(</mo><mi>F</mi><mo>)</mo></math> denote the inducibility of F and let <math><msub><mrow><mi>i</mi></mrow><mrow><mi>H</mi></mrow></msub><mo>(</mo><mi>F</mi><mo>)</mo></math> denote the inducibility of F over H-free graphs. We prove that for almost all graphs F on a given number of vertices, <math><msub><mrow><mi>i</mi></mrow><mrow><msub><mrow><mi>K</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></msub><mo>(</mo><mi>F</mi><mo>)</mo></math> attains infinitely many values as k varies. For complete partite graphs F (and, more generally, for symmetrizable families of graphs F), we prove that <math><msub><mrow><mi>i</mi></mrow><mrow><mi>H</mi></mrow></msub><mo>(</mo><mi>F</mi><mo>)</mo><mo>=</mo><msub><mrow><mi>i</mi></mrow><mrow><msub><mrow><mi>K</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></msub><mo>(</mo><mi>F</mi><mo>)</mo></math> where <math><mi>k</mi><mo>=</mo><mi>χ</mi><mo>(</mo><mi>H</mi><mo>)</mo></math>, and is attained by a complete ℓ-partite graphon <math><msub><mrow><mi>W</mi></mrow><mrow><mi>F</mi><mo>,</mo><mi>k</mi></mrow></msub></math>, where <math><mi>ℓ</mi><mo><</mo><mi>k</mi></math>.</div><div>We determine the part sizes of <math><msub><mrow><mi>W</mi></mrow><mrow><mi>F</mi><mo>,</mo><mi>k</mi></mrow></msub></math> for all k, whence determine <math><mi>i</mi><mo>(</mo><mi>F</mi><mo>)</mo></math>, whenever F is the Turán graph on s vertices and r parts, for all <math><mi>s</mi><mo>≤</mo><mn>3</mn><mi>r</mi><mo>+</mo><mn>1</mn></math>, which was recently proved by Liu, Mubayi, and Reiher for <math><mi>s</mi><mo>=</mo><mi>r</mi><mo>+</mo><mn>1</mn></math>. As a corollary, this determines the inducibility of all Turán graphs on at most 14 vertices. Furthermore, since inducibility is invariant under complement, this determines the inducibility of all matchings and, more generally, all graphs with maximum degree 1, of any size. Similarly, this determines the inducibility of all triangle factors, of any size.</div><div>For complete partite graphs F with at most one singleton part, we prove that <math><msub><mrow><mi>i</mi></mrow><mrow><msub><mrow><mi>K</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></msub><mo>(</mo><mi>F</mi><mo>)</mo></math> only attains finitely many values as k varies; in particular, there exists <math><mi>t</mi><mo>=</mo><mi>t</mi><mo>(</mo><mi>F</mi><mo>)</mo></math> such that <math><mi>i</mi><mo>(</mo><mi>F</mi><mo>)</mo></math> is attained by some complete t-partite graphon. This is best possible as it was shown by Liu, Pikhurko, Sharifzadeh, and Staden that this is not necessarily true if there are two singleton parts.</div>

对于图F和图H，设i(F)表示F的可诱导性，设iH(F)表示F在无H图上的可诱导性。我们证明了在给定顶点数的几乎所有图F上，iKk(F)随着k的变化获得无限多个值。对于完全部图F（以及更一般地说，对于图的可对称族F），我们证明了iH(F)=iKk(F)，其中k=χ(H)，并且可以由一个完全的_部图WF，k得到，其中_ <；k。我们确定了WF的零件尺寸，k对于所有k，由此确定i(F)，当F是s≤3r+1的s个顶点和r个零件上的Turán图，最近由Liu， Mubayi和Reiher证明了s=r+1。作为推论，这决定了所有Turán图在最多14个顶点上的可归纳性。此外，由于可归纳性在补下是不变的，这决定了所有匹配的可归纳性，更一般地说，决定了任何大小的所有最大度为1的图的可归纳性。同样，这决定了所有三角形因子的可归纳性，无论大小。对于最多有一个单元素部分的完全部图F，我们证明了iKk(F)在k变化时只能得到有限多个值；特别地，存在t=t(F)使得i(F)可以由某个完备的t部图得到。这是最好的可能，因为Liu， Pikhurko， Sharifzadeh和Staden已经证明，如果有两个单元素部分，这并不一定是正确的。最后，对于每一个r，我们给出了一个非平凡的充分条件，使一个完全的r部图F具有i(F)是由一个所有部分大小不同的完全部图得到的性质。

引用次数: 0

3-colorable planar graphs have an intersection segment representation using 3 slopes 三色平面图形具有使用3个斜率的相交段表示

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-11-28 DOI: 10.1016/j.jctb.2025.11.005

Daniel Gonçalves

In his PhD Thesis E.R. Scheinerman conjectured that planar graphs are intersection graphs of line segments in the plane. This conjecture was proved with two different approaches by J. Chalopin and the author, and by the author, L. Isenmann, and C. Pennarun. In the case of 3-colorable planar graphs E.R. Scheinerman conjectured that it is possible to restrict the set of slopes used by the segments to only 3 slopes. Here we prove this conjecture by using an approach introduced by S. Felsner to deal with contact representations of planar graphs with homothetic triangles.

Scheinerman在其博士论文中推测，平面图是平面上线段的交点图。这个猜想由J. Chalopin和作者，以及作者L. Isenmann和C. Pennarun用两种不同的方法证明。对于三色平面图形，E.R. Scheinerman推测，可以将线段使用的斜率集限制为仅3个斜率。本文利用S. Felsner提出的一种方法来处理具有同质三角形的平面图的接触表示，证明了这一猜想。

引用次数: 0

Intersecting families with covering number 3 与3号覆盖物相交的家族

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-11-24 DOI: 10.1016/j.jctb.2025.11.004

Andrey Kupavskii

The covering number of a family is the size of the smallest set that intersects all sets from the family. In 1978 Frankl determined for

n \geq n_{0} (k)

the largest intersecting family of k-element subsets of

[n]

with covering number 3. In this paper, we essentially settle this problem, showing that the same family is extremal for any

k \geq 100

and

n > 2 k

一个族的覆盖数是与该族所有集合相交的最小集合的大小。1978年Frankl在n≥n0(k)时确定了覆盖数为3的[n]的k元素子集的最大相交族。在本文中，我们基本上解决了这个问题，证明了对于任意k≥100和n>；2k，同一族是极值的。

引用次数: 0

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