Journal of Combinatorial Theory Series B最新文献

英文中文

Rigidity and reconstruction in matroids of highly connected graphs 高连通图的拟阵的刚性与重构

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2026-05-01 Epub Date: 2026-01-16 DOI: 10.1016/j.jctb.2026.01.003

Dániel Garamvölgyi

A graph matroid family

M

is a family of matroids

M (G)

defined on the edge set of each finite graph G in a compatible and isomorphism-invariant way. We say that

M

has the Whitney property if there is a constant c such that every c-connected graph G is uniquely determined by

M (G)

. Similarly,

M

has the Lovász-Yemini property if there is a constant c such that for every c-connected graph G,

M (G)

has maximal rank among graphs on the same number of vertices.

We show that if

M

is unbounded (that is, there is no absolute constant bounding the rank of

M (G)

for every G), then

M

has the Whitney property if and only if it has the Lovász-Yemini property. We also give a complete characterization of these properties in the bounded case. As an application, we show that if some graph matroid families have the Whitney property, then so does their union. Finally, we show that every 1-extendable graph matroid family has the Lovász-Yemini (and thus the Whitney) property. These results unify and extend a number of earlier results about graph reconstruction from an underlying matroid.

图拟阵族M是在每一个有限图G的边集中以相容且同构不变的方式定义的拟阵族M(G)。我们说M具有惠特尼性质，如果存在一个常数c使得每个c连通图G唯一地由M(G)决定。类似地，如果存在一个常数c，使得对于每个c连通图G， M(G)在具有相同顶点数的图中具有最大秩，则M具有Lovász-Yemini性质。我们证明，如果M是无界的（即，对于每个G， M(G)的秩没有绝对常数的边界），那么M具有惠特尼性质当且仅当它具有Lovász-Yemini性质。在有界情况下，我们也给出了这些性质的完整刻画。作为一个应用，我们证明了如果一些图阵族具有惠特尼性质，那么它们的并集也具有惠特尼性质。最后，我们证明了每一个1-可扩展图矩阵族都具有Lovász-Yemini（因此也是Whitney）性质。这些结果统一并扩展了先前关于从底层矩阵重构图的一些结果。

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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 : 2026-05-01 Epub 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

A lower bound on the number of edges in DP-critical graphs dp临界图中边数的下界

IF 1.4 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2026-03-20 DOI: 10.1016/j.jctb.2026.03.002

Peter Bradshaw, Ilkyoo Choi, Alexandr Kostochka, Jingwei Xu

引用次数: 0

Chords in longest cycles in 3-connected graphs 三连通图中最长循环的和弦

IF 1.4 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2026-03-19 DOI: 10.1016/j.jctb.2026.03.001

Carsten Thomassen

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 : 2026-03-01 Epub 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

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 : 2026-03-01 Epub 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

The Maker-Breaker percolation game on the square lattice 方块格子上的Maker-Breaker渗透游戏

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2026-03-01 Epub Date: 2025-11-11 DOI: 10.1016/j.jctb.2025.10.010

Vojtěch Dvořák, Adva Mond, Victor Souza

We study the

(m, b)

Maker-Breaker percolation game on

Z^{2}

, introduced by Day and Falgas-Ravry. In this game, on each of their turns, Maker and Breaker claim respectively m and b unclaimed edges of the square lattice

Z^{2}

. Breaker wins if the component containing the origin becomes finite when his edges are deleted from

Z^{2}

. Maker wins if she can indefinitely avoid a win of Breaker. We show that Breaker has a winning strategy for the

(m, b)

game whenever

b ⩾ (2 - 1 / 14 + o (1)) m

, breaking the ratio 2 barrier proved by Day and Falgas-Ravry.

Addressing further questions of Day and Falgas-Ravry, we show that Breaker can win the

(m, 2 m)

game even if he allows Maker to claim c edges before the game starts, for any integer c, and that he can moreover win rather quickly as a function of c.

We also consider the game played on the so-called polluted board, obtained after performing Bernoulli bond percolation on

Z^{2}

with parameter p. We show that for the

(1, 1)

game on the polluted board, Breaker almost surely has a winning strategy whenever

p ⩽ 0.6298

我们研究了Day和Falgas-Ravry在Z2上引入的（m,b） Maker-Breaker渗透博弈。在这个游戏中，制造者和破坏者在各自的回合中分别占有方形格子Z2的m条和b条无人认领的边。如果包含原点的组件在其边缘从Z2中删除时变得有限，则断路器获胜。如果制造者可以无限期地避免毁灭者的胜利，她就赢了。我们表明，每当b大于或等于（2−1/14+o(1)）m时，Breaker具有（m,b）游戏的制胜策略，打破了Day和Falgas-Ravry证明的比率2障碍。为了进一步解决Day和Falgas-Ravry的问题，我们证明了对于任何整数c，即使Breaker允许Maker在游戏开始前声称c条边，他也可以赢得（m,2m）游戏，并且作为c的函数，他还可以相当快地获胜。我们还考虑了在所谓的污染棋盘上进行的游戏，该游戏是在参数p的Z2上执行伯努利键渗透后获得的。我们表明，对于污染棋盘上的（1,1）游戏，当p≤0.6298时，破局者几乎肯定有一个获胜策略。

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引用次数: 0

Bisection width, discrepancy, and eigenvalues of hypergraphs 超图的二分宽度、差异和特征值

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2026-03-01 Epub Date: 2025-11-20 DOI: 10.1016/j.jctb.2025.11.003

Eero Räty , István Tomon

A celebrated result of Alon from 1993 states that any d-regular graph on n vertices (where

d = O (n^{1 / 9})

) has a bisection with at most

\frac{d n}{2} (\frac{1}{2} - Ω (\frac{1}{\sqrt{d}}))

edges, and this is optimal. Recently, this result was greatly extended by Räty, Sudakov, and Tomon. We build on the ideas of the latter, and use a semidefinite programming inspired approach to prove the following variant for hypergraphs: every r-uniform d-regular hypergraph on n vertices (where

d ≪ n^{1 / 2}

) has a bisection of size at most

\frac{d n}{r} (1 - \frac{1}{2^{r - 1}} - \frac{c}{\sqrt{d}}),

for some

c = c (r) > 0

. This bound is the best possible up to the precise value of c. Moreover, a bisection achieving this bound can be found by a polynomial-time randomized algorithm.

The minimum bisection is closely related to discrepancy. We also prove sharp bounds on the discrepancy and so called positive discrepancy of hypergraphs, extending results of Bollobás and Scott. Furthermore, we discuss implications about Alon-Boppana type bounds. We show that if H is an r-uniform d-regular hypergraph, then certain notions of second largest eigenvalue

λ_{2}

associated with the adjacency tensor satisfy

λ_{2} \geq Ω_{r} (\sqrt{d})

, improving results of Li and Mohar.

Alon在1993年的一个著名的结果表明，任何有n个顶点的d正则图（其中d=O(n1/9)）的等分最多有dn2（12−Ω（1d））条边，这是最优的。最近，Räty、Sudakov和Tomon极大地扩展了这一结果。我们以后者的思想为基础，使用半定规划启发的方法来证明超图的以下变体：在n个顶点（其中d≪n1/2）上的每个r-均匀d-正则超图的等分大小最多为dnr(1 - 12r - 1 - cd)，对于某些c=c(r)>0。这个边界是c精确值的最佳可能。此外，可以通过多项式时间随机化算法找到达到这个边界的平分。最小二分率与差值密切相关。我们还证明了超图的差值和所谓的正差值的尖锐界限，推广了Bollobás和Scott的结果。此外，我们讨论了关于Alon-Boppana型界的启示。我们证明了如果H是一个r-一致d-正则超图，那么与邻接张量相关的第二大特征值λ2的某些概念满足λ2≥Ωr(d)，改进了Li和Mohar的结果。

{"title":"Bisection width, discrepancy, and eigenvalues of hypergraphs","authors":"Eero Räty , István Tomon","doi":"10.1016/j.jctb.2025.11.003","DOIUrl":"10.1016/j.jctb.2025.11.003","url":null,"abstract":"<div><div>A celebrated result of Alon from 1993 states that any d-regular graph on n vertices (where <math><mi>d</mi><mo>=</mo><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>1</mn><mo>/</mo><mn>9</mn></mrow></msup><mo>)</mo></math>) has a bisection with at most <math><mfrac><mrow><mi>d</mi><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>(</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>−</mo><mi>Ω</mi><mo>(</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msqrt><mrow><mi>d</mi></mrow></msqrt></mrow></mfrac><mo>)</mo><mo>)</mo></math> edges, and this is optimal. Recently, this result was greatly extended by Räty, Sudakov, and Tomon. We build on the ideas of the latter, and use a semidefinite programming inspired approach to prove the following variant for hypergraphs: every r-uniform d-regular hypergraph on n vertices (where <math><mi>d</mi><mo>≪</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></math>) has a bisection of size at most<math><mfrac><mrow><mi>d</mi><mi>n</mi></mrow><mrow><mi>r</mi></mrow></mfrac><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>r</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow></mfrac><mo>−</mo><mfrac><mrow><mi>c</mi></mrow><mrow><msqrt><mrow><mi>d</mi></mrow></msqrt></mrow></mfrac><mo>)</mo></mrow><mo>,</mo></math> for some <math><mi>c</mi><mo>=</mo><mi>c</mi><mo>(</mo><mi>r</mi><mo>)</mo><mo>></mo><mn>0</mn></math>. This bound is the best possible up to the precise value of c. Moreover, a bisection achieving this bound can be found by a polynomial-time randomized algorithm.</div><div>The minimum bisection is closely related to discrepancy. We also prove sharp bounds on the discrepancy and so called positive discrepancy of hypergraphs, extending results of Bollobás and Scott. Furthermore, we discuss implications about Alon-Boppana type bounds. We show that if H is an r-uniform d-regular hypergraph, then certain notions of second largest eigenvalue <math><msub><mrow><mi>λ</mi></mrow><mrow><mn>2</mn></mrow></msub></math> associated with the adjacency tensor satisfy <math><msub><mrow><mi>λ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>≥</mo><msub><mrow><mi>Ω</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>(</mo><msqrt><mrow><mi>d</mi></mrow></msqrt><mo>)</mo></math>, improving results of Li and Mohar.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"177 ","pages":"Pages 186-215"},"PeriodicalIF":1.2,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145559890","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

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 : 2026-03-01 Epub 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

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 : 2026-03-01 Epub 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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