Journal of Combinatorial Theory Series B最新文献

英文中文

Non-degenerate hypergraphs with exponentially many extremal constructions 具有指数多极值结构的非退化超图

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-06-13 DOI: 10.1016/j.jctb.2025.06.001

József Balogh , Felix Christian Clemen , Haoran Luo

For every integer

t ⩾ 0

, denote by

F_{5}^{t}

the hypergraph on vertex set

{1, 2, \dots, 5 + t}

with hyperedges

{123, 124} \cup {34 k : 5 ⩽ k ⩽ 5 + t}

. We determine

ex (n, F_{5}^{t})

for every

t ⩾ 0

and sufficiently large n and characterize the extremal

F_{5}^{t}

-free hypergraphs. In particular, if n satisfies certain divisibility conditions, then the extremal

F_{5}^{t}

-free hypergraphs are exactly the balanced complete tripartite hypergraphs with additional hyperedges inside each of the three parts

(V_{1}, V_{2}, V_{3})

in the partition; each part

V_{i}

spans a

(| V_{i} |, 3, 2, t)

-design. This generalizes earlier work of Frankl and Füredi on the Turán number of

F_{5} : = F_{5}^{0}

Our results extend a theory of Erdős and Simonovits about the extremal constructions for certain fixed graphs. In particular, the hypergraphs

F_{5}^{6 t}

, for

t ⩾ 1

, are the first examples of hypergraphs with exponentially many extremal constructions and positive Turán density.

对于每个整数t小于0，用F5t表示顶点集{1,2，…，5+t}上的超图，超边{123,124}∪{34k:5≤k≤5+t}。我们为每个t小于0和足够大的n确定ex(n,F5t)，并表征极端无F5t超图。特别地，如果n满足一定的可除性条件，则无f5t极值超图就是分区中三个部分（V1,V2,V3）内各有附加超边的平衡完全三部超图；每个部分Vi跨越一个(|Vi|,3,2,t)-设计。这概括了Frankl和f redi关于F5:=F50的Turán数的早期工作。我们的结果推广了Erdős和Simonovits关于某些固定图的极值结构的理论。特别是，对于t大于或等于1的超图F56t，是具有指数级许多极值结构和正Turán密度的超图的第一个例子。

引用次数: 0

On the Keevash-Knox-Mycroft conjecture 基瓦什-诺克斯-麦考夫猜想

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-05-28 DOI: 10.1016/j.jctb.2025.05.003

Luyining Gan , Jie Han

Given

1 \leq ℓ < k

and

δ \geq 0

, let

PM (k, ℓ, δ)

be the decision problem for the existence of perfect matchings in n-vertex k-uniform hypergraphs with minimum ℓ-degree at least

δ (\begin{matrix} n - ℓ \\ k - ℓ \end{matrix})

. For

k \geq 3

PM (k, ℓ, 0)

was one of the first NP-complete problems by Karp. Keevash, Knox and Mycroft conjectured that

PM (k, ℓ, δ)

is in P for every

δ > 1 - {(1 - 1 / k)}^{k - ℓ}

and verified the case

ℓ = k - 1

In this paper we show that this problem can be reduced to the study of the minimum ℓ-degree condition forcing the existence of fractional perfect matchings. Together with existing results on fractional perfect matchings, this solves the conjecture of Keevash, Knox and Mycroft for

ℓ \geq 0.4 k

. Moreover, we also supply an algorithm that outputs a perfect matching, provided that one exists.

给定1≤l <；k,δ≥0，设PM（k, l,δ）为最小n-度且至少δ(n−k−l)的n顶点k-一致超图中存在完美匹配的判定问题。当k≥3时，PM（k, r,0）是Karp的第一个np完全问题。Keevash， Knox和Mycroft推测，对于每一个δ>；1−(1−1/k)k−r， PM（k, r, r）都在P中，并验证了r =k−1的情况。在本文中，我们证明了这个问题可以简化为对迫使分数阶完美匹配存在的最小阶条件的研究。结合已有的分数阶完美匹配结果，解决了Keevash， Knox和Mycroft对于r≥0.4k的猜想。此外，我们还提供了一个算法，如果存在完美匹配，则输出完美匹配。

引用次数: 0

The 1-2 conjecture holds for regular graphs 1-2猜想适用于正则图

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-05-19 DOI: 10.1016/j.jctb.2025.05.002

Kecai Deng , Hongyuan Qiu

The 1-2 conjecture asserts that the vertices and edges of every graph can be assigned with weights in

{1, 2}

such that adjacent vertices receive distinct weighted degrees. While this conjecture remains open in general, it has been proven that it is possible to achieve this using the weight set

{1, 2, 3}

. We demonstrate that the weight set

{0, 1}

suffices for every graph. As a corollary, the 1-2 conjecture is confirmed for regular graphs. Additionally, we verify another related conjecture concerning locally irregular total colouring, for regular graphs.

1-2猜想断言每个图的顶点和边都可以在{1,2}中赋予权重，使得相邻的顶点得到不同的加权度。虽然这个猜想在一般情况下仍然是开放的，但已经证明可以使用权值集{1,2,3}来实现这一点。我们证明了权值集{0,1}对每个图都是足够的。作为一个推论，对正则图证实了1-2猜想。此外，对于正则图，我们验证了另一个关于局部不规则全着色的相关猜想。

引用次数: 0

Connectivity keeping paths containing prescribed vertices in highly connected triangle-free graphs 在高度连通的无三角形图中保持包含规定顶点的路径的连通性

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-05-16 DOI: 10.1016/j.jctb.2025.05.001

Shinya Fujita

Let

m, k

be integers with

m \geq 1, k \geq 2

. For a k-connected graph G, a subgraph H of G is k-removable if

G - V (H)

is still a k-connected graph. A graph is triangle-free if it contains no triangle as a subgraph.

In this paper, we prove that if G is a k-connected triangle-free graph with minimum degree at least

k + (m - 1) / 2

, then for any vertex

v \in V (G)

, there exists a path P on m vertices starting from v such that

G - V (P)

is a

(k - 1)

-connected graph. This result is obtained by showing a stronger statement concerning the existence of k-removable paths in k-connected triangle-free graphs. We also prove that if G is a k-connected triangle-free graph with minimum degree at least

k + 1

, then G contains a k-removable edge. Our results confirm a conjecture due to Luo et al. concerning the existence of a k-removable path on m vertices in a k-connected bipartite graph for all odd m together with the case

m = 2

设m，k为m≥1，k≥2的整数。对于k连通图G，如果G−V(H)仍然是k连通图，则G的子图H是k可移动的。如果一个图的子图中不包含三角形，那么这个图就是无三角形的。证明了如果G是一个最小度至少为k+(m−1)/2的k连通无三角形图，那么对于任意顶点v∈v (G)，在从v出发的m个顶点上存在一条路径P，使得G−v (P)是一个（k−1）连通图。这一结果是通过证明k连通无三角形图中k条可移动路径的存在性而得到的。我们还证明了如果G是一个k连通且最小度至少为k+1的无三角形图，则G包含一条k可移动边。我们的结果证实了Luo等人关于k连通二部图的m个顶点上存在k个可移动路径的猜想，对于所有奇数m以及m=2的情况。

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引用次数: 0

A structure theorem for pseudosegments and its applications 伪段的一个结构定理及其应用

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-05-06 DOI: 10.1016/j.jctb.2025.04.007

Jacob Fox , János Pach , Andrew Suk

We prove a far-reaching strengthening of Szemerédi's regularity lemma for intersection graphs of pseudosegments. It shows that the vertex set of such a graph can be partitioned into a bounded number of parts of roughly the same size such that almost all bipartite graphs between different pairs of parts are complete or empty. We use this to get an improved bound on disjoint edges in simple topological graphs, showing that every n-vertex simple topological graph with no k pairwise disjoint edges has at most

n {(\log n)}^{O (\log k)}

edges.

我们证明了伪段相交图的szemersamudi正则引理的一个意义深远的强化。证明了这种图的顶点集可以被分割成有限个大小大致相同的部分，使得不同部分对之间的二部图几乎都是完全的或空的。我们用它得到了简单拓扑图中不相交边的改进界，证明了每一个没有k对不相交边的n顶点简单拓扑图最多有n(log log n)O（log k）条边。

引用次数: 0

A characterization of testable hypergraph properties 可测试超图性质的表征

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-05-06 DOI: 10.1016/j.jctb.2025.04.009

Felix Joos , Jaehoon Kim , Daniela Kühn , Deryk Osthus

We provide a combinatorial characterization of all testable properties of k-uniform hypergraphs (k-graphs for short). Here, a k-graph property P is testable if there is a randomized algorithm which makes a bounded number of edge queries and distinguishes with probability 2/3 between k-graphs that satisfy P and those that are far from satisfying P. For the 2-graph case, such a combinatorial characterization was obtained by Alon, Fischer, Newman and Shapira. Our results for the k-graph setting are in contrast to those of Austin and Tao, who showed that for the somewhat stronger concept of local repairability, the testability results for graphs do not extend to the 3-graph setting.

我们提供了k-一致超图（简称k图）的所有可测试性质的组合表征。这里，如果存在一种随机算法，该算法进行有限数量的边查询，并以2/3的概率区分满足P的k图和远远不满足P的k图，则k图性质P是可测试的。对于2图情况，Alon, Fischer， Newman和Shapira获得了这样的组合表征。我们对k图设置的结果与Austin和Tao的结果相反，他们表明，对于更强的局部可修复性概念，图的可测试性结果不能扩展到3图设置。

引用次数: 0

Finding irregular subgraphs via local adjustments 通过局部调整查找不规则子图

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-05-02 DOI: 10.1016/j.jctb.2025.04.008

Jie Ma , Shengjie Xie

For a graph H, let

m (H, k)

denote the number of vertices of degree k in H. A conjecture of Alon and Wei states that for any

d \geq 3

, every n-vertex d-regular graph contains a spanning subgraph H satisfying

| m (H, k) - \frac{n}{d + 1} | \leq 2

for every

0 \leq k \leq d

. This holds easily when

d \leq 2

. An asymptotic version of this conjecture was initially established by Frieze, Gould, Karoński and Pfender, subsequently improved by Alon and Wei, and most recently enhanced by Fox, Luo and Pham, approaching its complete range. All of these approaches relied on probabilistic methods.

In this paper, we provide a novel framework to study this conjecture, based on localized deterministic techniques which we call local adjustments. We prove two main results. Firstly, we show that every n-vertex d-regular graph contains a spanning subgraph H satisfying

| m (H, k) - \frac{n}{d + 1} | \leq 2 d^{2}

for all

0 \leq k \leq d

, which provides the first bound independent of the value of n. Secondly, we confirm the case

d = 3

of the Alon-Wei Conjecture in a strong form. Both results can be generalized to multigraphs and yield efficient algorithms for finding the desired subgraphs H. Furthermore, we explore a generalization of the Alon-Wei Conjecture for multigraphs and its connection to the Faudree-Lehel Conjecture concerning irregularity strength.

对于图H，设m（H,k）表示H中k度顶点的个数。Alon和Wei的一个猜想指出，对于任意d≥3，每个n顶点的d正则图都包含一个生成子图H，满足|m(H,k)−和+1|≤2，对于每0≤k≤d。当d≤2时，这很容易成立。这个猜想的渐近版本最初由Frieze， Gould， Karoński和Pfender建立，随后由Alon和Wei改进，最近由Fox， Luo和Pham加强，接近其完整范围。所有这些方法都依赖于概率方法。在本文中，我们提供了一个新的框架来研究这一猜想，基于局部确定性技术，我们称之为局部调整。我们证明了两个主要结果。首先，我们证明了每个n顶点d正则图都包含一个生成子图H满足|m(H,k)−和+1|≤2d2，这提供了与n值无关的第一界。其次，我们以强形式证实了Alon-Wei猜想d=3的情况。这两个结果都可以推广到多图中，并给出了寻找所需子图h的有效算法。此外，我们探讨了多图的Alon-Wei猜想的推广及其与关于不规则强度的Faudree-Lehel猜想的联系。

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引用次数: 0

Weak diameter choosability of graphs with an excluded minor 具有排除次要项的图的弱直径可选择性

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-04-23 DOI: 10.1016/j.jctb.2025.04.005

Joshua Crouch, Chun-Hung Liu

Weak diameter coloring of graphs recently attracted attention, partially due to its connection to asymptotic dimension of metric spaces. We consider weak diameter list-coloring of graphs in this paper. Dvořák and Norin proved that graphs with bounded Euler genus are 3-choosable with bounded weak diameter. In this paper, we extend their result by showing that for every graph H, H-minor free graphs are 3-choosable with bounded weak diameter. The upper bound 3 is optimal and it strengthens an earlier result for non-list-coloring H-minor free graphs with bounded weak diameter. As a corollary, H-minor free graphs with bounded maximum degree are 3-choosable with bounded clustering, strengthening an earlier result for non-list-coloring.

When H is planar, we prove a much stronger result: for every 2-list-assignment L of an H-minor free graph, every precoloring with bounded weak diameter can be extended to an L-coloring with bounded weak diameter. It is a common generalization of earlier results for non-list-coloring with bounded weak diameter and for list-coloring with bounded clustering without allowing precolorings. As a corollary, for any planar graph H and H-minor free graph G, there are exponentially many list-colorings of G with bounded weak diameter (and with bounded clustering if G also has bounded maximum degree); and every graph with bounded layered tree-width and bounded maximum degree has exponentially many 3-colorings with bounded clustering.

We also show that the aforementioned results for list-coloring cannot be extended to odd minor free graphs by showing that some bipartite graphs with maximum degree Δ are k-choosable with bounded weak diameter only when

k = Ω (\log Δ / \log \log Δ)

. On the other hand, we show that odd H-minor graphs are 3-colorable with bounded weak diameter, implying an earlier result about clustered coloring of odd H-minor free graphs with bounded maximum degree.

图的弱直径着色近年来引起人们的关注，部分原因是它与度量空间的渐近维数有关。本文研究了图的弱直径表着色问题。Dvořák和Norin证明了具有有界欧拉属的图是具有有界弱直径的3-可选图。在本文中，我们推广了它们的结果，证明了对于每一个图H， H次自由图都是具有有界弱直径的3-可选图。上界3是最优的，它加强了之前关于弱直径有界的非列表着色h次自由图的结果。作为一个推论，具有有界最大度的h次自由图在有界聚类中是3-可选的，加强了之前关于非列表着色的结果。当H是平面时，我们证明了一个更强的结果：对于H次自由图的每一个2-列表赋值L，每一个弱直径有界的预着色都可以推广到弱直径有界的L着色。对于有界弱直径的非列表着色和不允许预着色的有界聚类的列表着色，这是早期结果的一般推广。作为推论，对于任意平面图H和H次自由图G， G的弱直径有界（如果G的最大度也有界，则G的聚类有界）存在指数多列着色；并且每一个层树宽度有界、最大度有界的图都有指数次的有界聚类的3色。通过证明一些最大度为Δ的二部图只有在k=Ω（log （Δ） /log (log)）时才具有弱直径有界的k-可选性，我们还证明了上述关于列表着色的结果不能推广到奇次自由图。另一方面，我们证明了奇h小图是弱直径有界的3色图，暗示了关于最大度有界的奇h小自由图的聚类着色的早期结果。

{"title":"Weak diameter choosability of graphs with an excluded minor","authors":"Joshua Crouch, Chun-Hung Liu","doi":"10.1016/j.jctb.2025.04.005","DOIUrl":"10.1016/j.jctb.2025.04.005","url":null,"abstract":"<div><div>Weak diameter coloring of graphs recently attracted attention, partially due to its connection to asymptotic dimension of metric spaces. We consider weak diameter list-coloring of graphs in this paper. Dvořák and Norin proved that graphs with bounded Euler genus are 3-choosable with bounded weak diameter. In this paper, we extend their result by showing that for every graph H, H-minor free graphs are 3-choosable with bounded weak diameter. The upper bound 3 is optimal and it strengthens an earlier result for non-list-coloring H-minor free graphs with bounded weak diameter. As a corollary, H-minor free graphs with bounded maximum degree are 3-choosable with bounded clustering, strengthening an earlier result for non-list-coloring.</div><div>When H is planar, we prove a much stronger result: for every 2-list-assignment L of an H-minor free graph, every precoloring with bounded weak diameter can be extended to an L-coloring with bounded weak diameter. It is a common generalization of earlier results for non-list-coloring with bounded weak diameter and for list-coloring with bounded clustering without allowing precolorings. As a corollary, for any planar graph H and H-minor free graph G, there are exponentially many list-colorings of G with bounded weak diameter (and with bounded clustering if G also has bounded maximum degree); and every graph with bounded layered tree-width and bounded maximum degree has exponentially many 3-colorings with bounded clustering.</div><div>We also show that the aforementioned results for list-coloring cannot be extended to odd minor free graphs by showing that some bipartite graphs with maximum degree Δ are k-choosable with bounded weak diameter only when <math><mi>k</mi><mo>=</mo><mi>Ω</mi><mo>(</mo><mi>log</mi><mo>⁡</mo><mi>Δ</mi><mo>/</mo><mi>log</mi><mo>⁡</mo><mi>log</mi><mo>⁡</mo><mi>Δ</mi><mo>)</mo></math>. On the other hand, we show that odd H-minor graphs are 3-colorable with bounded weak diameter, implying an earlier result about clustered coloring of odd H-minor free graphs with bounded maximum degree.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"174 ","pages":"Pages 28-70"},"PeriodicalIF":1.2,"publicationDate":"2025-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143864224","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

Induced C4-free subgraphs with large average degree 具有较大平均度的无 C4 子图

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-04-23 DOI: 10.1016/j.jctb.2025.04.002

Xiying Du , António Girão , Zach Hunter , Rose McCarty , Alex Scott

We prove that there exists a constant C so that, for all

s, k \in N

, if G has average degree at least

k^{C s^{3}}

and does not contain

K_{s, s}

as a subgraph then it contains an induced subgraph which is

C_{4}

-free and has average degree at least k. It was known that some function of s and k suffices, but this is the first explicit bound. We give several applications of this result, including short and streamlined proofs of the following two corollaries.

We show that there exists a constant C so that, for all

s, k \in N

, if G has average degree at least

k^{C s^{3}}

and does not contain

K_{s, s}

as a subgraph then it contains an induced subdivision of

K_{k}

. This is the first quantitative improvement on a well-known theorem of Kühn and Osthus; their proof gives a bound that is triply exponential in both k and s.

We also show that for any hereditary degree-bounded class

F

, there exists a constant

C = C_{F}

so that

C^{s^{3}}

is a degree-bounding function for

F

. This is the first bound of any type on the rate of growth of such functions.

我们证明了存在一个常数C，使得对于所有s，k∈N，如果G的平均度至少为kCs3且不包含Ks，s作为子图，则它包含一个不含c4且平均度至少为k的诱导子图。已知s和k的某个函数是足够的，但这是第一个显式界。我们给出了这个结果的几个应用，包括以下两个推论的简短和简化的证明。我们证明了存在一个常数C，使得对于所有s，k∈N，如果G的平均度至少为kCs3，并且不包含Ks，s作为子图，则它包含Kk的诱导子图。这是对k hn和Osthus的一个著名定理的第一个定量改进；他们的证明给出了一个在k和s上都是三指数的界。我们还证明了对于任何遗传度有界类F，存在一个常数C=CF，使得Cs3是F的一个度有界函数。这是关于这类函数增长率的任何类型的第一个界。

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引用次数: 0

A matrix realization of spectral bounds 谱界的矩阵实现

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-04-23 DOI: 10.1016/j.jctb.2025.04.006

Yen-Jen Cheng , Chih-wen Weng

We give a unified and systematic way to find bounds for the largest real eigenvalue of a nonnegative matrix by considering its modified quotient matrix. We leverage this insight to identify the unique matrix whose largest real eigenvalue is maximum among all

(0, 1)

-matrices with a specified number of ones. This result resolves a problem that was posed independently by R. Brualdi and A. Hoffman, as well as F. Friedland, back in 1985.

利用非负矩阵的修正商矩阵，给出了求非负矩阵最大实特征值界的统一、系统的方法。我们利用这一见解来识别唯一矩阵，其最大实特征值在所有(0,1)-具有指定数量的矩阵中是最大的。这一结果解决了R. Brualdi和a . Hoffman以及F. Friedland在1985年独立提出的一个问题。

引用次数: 0

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