Journal of Combinatorial Theory Series B最新文献

英文中文

A problem of Erdős and Hajnal on paths with equal-degree endpoints 具有等度端点的路径上的Erdős和Hajnal问题

IF 1.4 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2026-02-05 DOI: 10.1016/j.jctb.2026.01.006

Kaizhe Chen, Jie Ma

引用次数: 0

Hom complexes of graphs whose codomains are square-free 图的上域是无平方的Hom复形

IF 1.4 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2026-02-02 DOI: 10.1016/j.jctb.2026.01.005

Takahiro Matsushita

引用次数: 0

Universality for graphs with bounded density 有界密度图的通用性

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2026-01-28 DOI: 10.1016/j.jctb.2026.01.004

Noga Alon , Natalie Dodson , Carmen Jackson , Rose McCarty , Rajko Nenadov , Lani Southern

A graph G is universal for a (finite) family

H

of graphs if every

H \in H

is a subgraph of G. For a given family

H

, the goal is to determine the smallest number of edges an

H

-universal graph can have. With the aim of unifying a number of recent results, we consider a family of graphs with bounded density. In particular, we construct a graph with

O_{d} (n^{2 - 1 / (⌈ d ⌉ + 1)})

edges which contains every n-vertex graph with density at most

d \in Q

(

d \geq 1

), which is close to a

Ω (n^{2 - 1 / d})

lower bound obtained by counting lifts of a carefully chosen (small) graph. When restricting the maximum degree of such graphs to be constant, we obtain near-optimal universality. If we further assume

d \in N

, we get an asymptotically optimal construction.

如果每个H∈H是G的子图，则图G对于图的（有限）族H是全称的。对于给定的族H，目标是确定H-全称图可以拥有的最小边数。为了统一最近的一些结果，我们考虑了一组有界密度的图。特别地，我们构造了一个有thod （n2−1/(≤d²+1)）条边的图，它包含了密度不超过d∈Q （d≥1）的每一个n顶点图，该图接近于一个Ω（n2−1/d）下界，该下界是通过对一个精心选择的（小）图的升力进行计数得到的。当将这类图的最大度限定为常数时，我们得到了近似最优通用性。如果我们进一步假设d∈N，我们得到一个渐近最优结构。

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引用次数: 0

Excluding disjoint Kuratowski graphs 排除不相交的Kuratowski图

IF 1.4 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2026-01-20 DOI: 10.1016/j.jctb.2026.01.002

Neil Robertson, Paul Seymour

引用次数: 0

Excluding sums of Kuratowski graphs 排除Kuratowski图的和

IF 1.4 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2026-01-19 DOI: 10.1016/j.jctb.2026.01.001

Neil Robertson, Paul Seymour

引用次数: 0

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 : 2026-01-16 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

Integral biflow maximization 积分流最大化

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2026-01-08 DOI: 10.1016/j.jctb.2025.12.006

Guoli Ding , Rongchuan Tao , Mengxi Yang , Wenan Zang

Let

G = (V, E)

be a graph with four distinguished vertices, two sources

s_{1}, s_{2}

and two sinks

t_{1}, t_{2}

, let

c : E \to Z_{+}

be a capacity function, and let

P

be the set of all simple paths in G from

s_{1}

t_{1}

or from

s_{2}

t_{2}

. A biflow (or 2-commodity flow) in G is an assignment

f : P \to R_{+}

such that

\sum_{e \in Q \in P} f (Q) \leq c (e)

for all

e \in E

, whose value is defined to be

\sum_{Q \in P} f (Q)

. A bicut in G is a subset K of E that contains at least one edge from each member of

P

, whose capacity is

\sum_{e \in K} c (e)

. In 1977 Seymour characterized, in terms of forbidden structures, all graphs G for which the max-biflow (integral) min-bicut theorem holds true (that is, the maximum value of an integral biflow is equal to the minimum capacity of a bicut for every capacity function c); such a graph G is referred to as a Seymour graph. Nevertheless, his proof is not algorithmic in nature. In this paper we present a combinatorial polynomial-time algorithm for finding maximum integral biflows in Seymour graphs, which relies heavily on a structural description of such graphs.

设G=（V，E）为具有四个不同顶点、两个源s1，s2和两个汇t1，t2的图，设c:E→Z+为容量函数，设P为G中从s1到t1或从s2到t2的所有简单路径的集合。G中的双商品流（或双商品流）是一个赋值f:P→R+，使得所有e∈e的∑e∈Q∈Pf(Q)≤c(e)，其值定义为∑Q∈Pf(Q)。G中的二分切割是E的一个子集K，它包含来自P中每个元素的至少一条边，其容量为∑E∈Kc(E)。1977年，Seymour用禁止结构刻画了所有图G，其中最大-分流（积分）最小-二分定理成立（即对于每一个容量函数c，积分分流的最大值等于二分的最小容量）；这样的图G称为西摩图。然而，他的证明本质上不是算法。本文提出了一种查找西摩图中最大积分分岔的组合多项式时间算法，该算法在很大程度上依赖于西摩图的结构描述。

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引用次数: 0

Strong log-convexity of genus sequences 属序列的强对数凸性

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2026-01-06 DOI: 10.1016/j.jctb.2025.12.005

Bojan Mohar

For a graph G, and a nonnegative integer g, let

a_{g} (G)

be the number of 2-cell embeddings of G in an orientable surface of genus g (counted up to the combinatorial homeomorphism equivalence). In 1989, Gross, Robbins, and Tucker [21] proposed a conjecture that the sequence

a_{0} (G), a_{1} (G), a_{2} (G), \dots

is log-concave for every graph G. This conjecture is reminiscent to the Heron–Rota–Welsh Log Concavity Conjecture that was recently resolved in the affirmative by June Huh et al. [24], [1], except that it is closer to the notion of Δ-matroids than to the usual matroids. In this short paper, we disprove the Log Concavity Conjecture of Gross, Robbins, and Tucker by providing examples that show strong deviation from log-concavity at multiple terms of their genus sequences.

对于图G和非负整数G，设ag(G)为G在G属的可定向曲面上的2细胞嵌入数（计数到组合同胚等价）。1989年，Gross， Robbins和Tucker[21]提出了一个猜想，即序列a0(G),a1(G),a2(G)，…对于每个图G都是对数凹的。这个猜想让人联想到Heron-Rota-Welsh Log凹凸猜想，该猜想最近被June Huh等人以肯定的方式解决了。[24],[1]，除了它更接近Δ-matroids的概念而不是通常的拟阵。在这篇简短的文章中，我们通过提供在其属序列的多个项上显示强烈偏离对数凹性的例子来证明Gross， Robbins和Tucker的对数凹性猜想是错误的。

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引用次数: 0

The maximum sum of sizes of non-empty pairwise cross-intersecting families 非空成对交叉族的最大大小和

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2026-01-02 DOI: 10.1016/j.jctb.2025.12.004

Yang Huang , Yuejian Peng , Jian Wang

<div><div>Two families <math><mi>A</mi></math> and <math><mi>B</mi></math> are cross-intersecting if <math><mi>A</mi><mo>∩</mo><mi>B</mi><mo>≠</mo><mo>∅</mo></math> for any <math><mi>A</mi><mo>∈</mo><mi>A</mi></math> and <math><mi>B</mi><mo>∈</mo><mi>B</mi></math>. We call t families <math><msub><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>t</mi></mrow></msub></math> pairwise cross-intersecting families if <math><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub></math> and <math><msub><mrow><mi>A</mi></mrow><mrow><mi>j</mi></mrow></msub></math> are cross-intersecting for <math><mn>1</mn><mo>≤</mo><mi>i</mi><mo><</mo><mi>j</mi><mo>≤</mo><mi>t</mi></math>. Additionally, if <math><msub><mrow><mi>A</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>≠</mo><mo>∅</mo></math> for each <math><mi>j</mi><mo>∈</mo><mo>[</mo><mi>t</mi><mo>]</mo></math>, then we say that <math><msub><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>t</mi></mrow></msub></math> are non-empty pairwise cross-intersecting. Let <math><msub><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>⊂</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow></mtd></mtr><mtr><mtd><msub><mrow><mi>k</mi></mrow><mrow><mn>1</mn></mrow></msub></mtd></mtr></mtable><mo>)</mo></mrow><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>⊂</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow></mtd></mtr><mtr><mtd><msub><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msub></mtd></mtr></mtable><mo>)</mo></mrow><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>⊂</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow></mtd></mtr><mtr><mtd><msub><mrow><mi>k</mi></mrow><mrow><mi>t</mi></mrow></msub></mtd></mtr></mtable><mo>)</mo></mrow></math> be non-empty pairwise cross-intersecting families with <math><mi>t</mi><mo>≥</mo><mn>2</mn></math>, <math><msub><mrow><mi>k</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>≥</mo><msub><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>≥</mo><mo>⋯</mo><mo>≥</mo><msub><mrow><mi>k</mi></mrow><mrow><mi>t</mi></mrow></msub></math>, and <math><mi>n</mi><mo>≥</mo><msub><mrow><mi>k</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msub></math>, we determine the maximum of <math><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>t</mi></mr

对于任意A∈A， B∈B，若A∩B≠∅，则两个族A与B相交。如果Ai和Aj在1≤i<；j≤t时相交，我们称t族为A1，A2，…，At成对相交族。另外，若Aj≠∅对于每个j∈[t]，则我们说A1，A2，…，At是非空的两两相交。设A1∧([n]k1),A2∧([n]k2)，…，At∧([n]kt)为t≥2，k1≥k2≥⋯≥kt， n≥k1+k2的非空两两相交族，我们确定∑i=1t|Ai|的最大值，并对所有极值族进行表征。这回答了Shi、Frankl和Qian [Combinatorica 42(2022)]提出的一个问题。

引用次数: 0

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

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