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 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2026-07-01 Epub Date: 2026-02-05 DOI: 10.1016/j.jctb.2026.01.006

Kaizhe Chen , Jie Ma

We address a problem posed by Erdős and Hajnal in 1991, proving that for all

n \geq 600

, every

(2 n + 1)

-vertex graph with at least

n^{2} + n + 1

edges contains two vertices of equal degree connected by a path of length three. The complete bipartite graph

K_{n, n + 1}

demonstrates that this edge bound is sharp. We further establish an analogous result for graphs with even order and investigate several related extremal problems.

我们解决了Erdős和Hajnal在1991年提出的一个问题，证明了对于所有n≥600，每个（2n+1）顶点图，至少有n2+n+1条边，包含两个等度的顶点，由长度为3的路径连接。完备二部图Kn，n+1证明了这个边界是尖锐的。我们进一步建立了偶阶图的类似结果，并研究了几个相关的极值问题。

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 : 2026-05-01 Epub 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

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 : 2026-05-01 Epub 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 : 2026-05-01 Epub 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

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 : 2026-05-01 Epub 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

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 : 2026-05-01 Epub 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

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 : 2026-05-01 Epub 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

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 : 2026-05-01 Epub 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。

{"title":"Integral biflow maximization","authors":"Guoli Ding , Rongchuan Tao , Mengxi Yang , Wenan Zang","doi":"10.1016/j.jctb.2025.12.006","DOIUrl":"10.1016/j.jctb.2025.12.006","url":null,"abstract":"<div><div>Let <math><mi>G</mi><mo>=</mo><mo>(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>)</mo></math> be a graph with four distinguished vertices, two sources <math><msub><mrow><mi>s</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>s</mi></mrow><mrow><mn>2</mn></mrow></msub></math> and two sinks <math><msub><mrow><mi>t</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>t</mi></mrow><mrow><mn>2</mn></mrow></msub></math>, let <math><mi>c</mi><mo>:</mo><mspace></mspace><mi>E</mi><mo>→</mo><msub><mrow><mi>Z</mi></mrow><mrow><mo>+</mo></mrow></msub></math> be a capacity function, and let <math><mi>P</mi></math> be the set of all simple paths in G from <math><msub><mrow><mi>s</mi></mrow><mrow><mn>1</mn></mrow></msub></math> to <math><msub><mrow><mi>t</mi></mrow><mrow><mn>1</mn></mrow></msub></math> or from <math><msub><mrow><mi>s</mi></mrow><mrow><mn>2</mn></mrow></msub></math> to <math><msub><mrow><mi>t</mi></mrow><mrow><mn>2</mn></mrow></msub></math>. A biflow (or 2-commodity flow) in G is an assignment <math><mi>f</mi><mo>:</mo><mspace></mspace><mi>P</mi><mo>→</mo><msub><mrow><mi>R</mi></mrow><mrow><mo>+</mo></mrow></msub></math> such that <math><msub><mrow><mo>∑</mo></mrow><mrow><mi>e</mi><mo>∈</mo><mi>Q</mi><mo>∈</mo><mi>P</mi></mrow></msub><mspace></mspace><mi>f</mi><mo>(</mo><mi>Q</mi><mo>)</mo><mo>≤</mo><mi>c</mi><mo>(</mo><mi>e</mi><mo>)</mo></math> for all <math><mi>e</mi><mo>∈</mo><mi>E</mi></math>, whose value is defined to be <math><msub><mrow><mo>∑</mo></mrow><mrow><mi>Q</mi><mo>∈</mo><mi>P</mi></mrow></msub><mspace></mspace><mi>f</mi><mo>(</mo><mi>Q</mi><mo>)</mo></math>. A bicut in G is a subset K of E that contains at least one edge from each member of <math><mi>P</mi></math>, whose capacity is <math><msub><mrow><mo>∑</mo></mrow><mrow><mi>e</mi><mo>∈</mo><mi>K</mi></mrow></msub><mspace></mspace><mi>c</mi><mo>(</mo><mi>e</mi><mo>)</mo></math>. 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.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"178 ","pages":"Pages 180-210"},"PeriodicalIF":1.2,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145925956","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

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 : 2026-05-01 Epub 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

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 : 2026-05-01 Epub 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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