Journal of Combinatorial Theory Series B最新文献

英文中文

Reuniting χ-boundedness with polynomial χ-boundedness 将χ-有界性与多项式χ-有界性重新统一

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-08-28 DOI: 10.1016/j.jctb.2025.08.002

Maria Chudnovsky , Linda Cook , James Davies , Sang-il Oum

A class

F

of graphs is χ-bounded if there is a function f such that

χ (H) \leq f (ω (H))

for all induced subgraphs H of a graph in

F

. If f can be chosen to be a polynomial, we say that

F

is polynomially χ-bounded. Esperet proposed a conjecture that every χ-bounded class of graphs is polynomially χ-bounded. This conjecture has been disproved; it has been shown that there are classes of graphs that are χ-bounded but not polynomially χ-bounded. Nevertheless, inspired by Esperet's conjecture, we introduce Pollyanna classes of graphs. A class

C

of graphs is Pollyanna if

C \cap F

is polynomially χ-bounded for every χ-bounded class

F

of graphs. We prove that several classes of graphs are Pollyanna and also present some proper classes of graphs that are not Pollyanna.

如果存在一个函数F，使得F中图的所有诱导子图H的χ(H)≤F (ω(H))，则一类图F是χ-有界的。如果F可以选择为多项式，则我们说F是多项式χ-有界的。Esperet提出了一个猜想，即每一类有χ有界的图都是多项式有χ有界的。这个猜想已经被推翻了；已经证明有一类图是χ有界的，但不是多项式χ有界的。然而，受Esperet猜想的启发，我们引入了波利安娜图类。如果C∩F对于每一个有χ有界的图类F都是多项式χ有界的，那么C类图就是波利安娜。我们证明了几类图是盲目乐观的，并给出了一些非盲目乐观图的适当类别。

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引用次数: 0

On graphs without cycles of length 0 modulo 4 在没有周期长度为0模4的图上

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-08-20 DOI: 10.1016/j.jctb.2025.07.008

Ervin Győri , Binlong Li , Nika Salia , Casey Tompkins , Kitti Varga , Manran Zhu

Bollobás proved that for every k and ℓ such that

k Z + ℓ

contains an even number, an n-vertex graph containing no cycle of length

ℓ \mod k

can contain at most a linear number of edges. The precise (or asymptotic) value of the maximum number of edges in such a graph is known for very few pairs ℓ and k. In this work we precisely determine the maximum number of edges in a graph containing no cycle of length

0 \mod 4

Bollobás证明了对于每一个k和r，使得k z + r包含一个偶数，一个n顶点的图，不包含长度为r modk的循环，最多只能包含一个线性数的边。这种图中最大边数的精确（或渐近）值对于很少的对（r和k）是已知的。在这项工作中，我们精确地确定了不包含长度为0mod4的循环的图中的最大边数。

引用次数: 0

Closure property of contraction-depth of matroids 拟阵收缩深度的闭包性

IF 1.4 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-08-08 DOI: 10.1016/j.jctb.2025.07.006

Marcin Briański, Daniel Král', Ander Lamaison

Contraction⁎-depth is a matroid depth parameter analogous to tree-depth of graphs. We establish the matroid analogue of the classical graph theory result asserting that the tree-depth of a graph G is the minimum height of a rooted forest whose closure contains G by proving the following for every matroid M (except the trivial case when M consists of loops and coloops only): the contraction⁎-depth of M plus one is equal to the minimum contraction-depth of a matroid containing M as a restriction.

收缩-depth是一个矩阵深度参数，类似于图的树深度。我们通过对每个矩阵M（除了M只由环和圈组成的平凡情况外）证明以下内容，建立了经典图论结果的拟阵模拟，该结果断言图G的树深是闭包包含G的根森林的最小高度：M + 1的收缩深度等于包含M作为约束的矩阵的最小收缩深度。

引用次数: 0

The codegree Turán density of 3-uniform tight cycles 3均匀紧循环的余度Turán密度

IF 1.4 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-08-07 DOI: 10.1016/j.jctb.2025.07.007

Simón Piga, Nicolás Sanhueza-Matamala, Mathias Schacht

Given any ε>0 we prove that every sufficiently large n-vertex 3-graph H where every pair of vertices is contained in at least (1/3+ε)n edges contains a copy of C10, i.e. the tight cycle on 10 vertices. In fact we obtain the same conclusion for every cycle Cℓ with ℓ≥19.

给定任意ε>；0，我们证明了每一个足够大的n顶点3-图H，其中每一对顶点至少包含（1/3+ε）n条边，其中包含C10的一个副本，即10个顶点上的紧环。事实上，对于每一个循环，我们都得到了相同的结论。

引用次数: 0

Stability in Bondy's theorem on paths and cycles 路径和循环上邦迪定理的稳定性

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-08-04 DOI: 10.1016/j.jctb.2025.07.004

Bo Ning , Long-Tu Yuan

In this paper, we study the stability result of a well-known theorem of Bondy. We prove that for any 2-connected non-hamiltonian graph, if every vertex except for at most one vertex has degree at least k, then it contains a cycle of length at least

2 k + 2

except for some special families of graphs. Our results imply several previous classical theorems including a deep and old result by Voss. We point out our result on stability in Bondy's theorem can directly imply a positive solution (in a slight stronger form) to the following problem: Is there a polynomial time algorithm to decide whether a 2-connected graph G on n vertices has a cycle of length at least

\min {2 δ (G) + 2, n}

? This problem originally motivates the recent study on algorithmic aspects of Dirac's theorem by Fomin, Golovach, Sagunov, and Simonov, although a stronger problem was solved by them by completely different methods. Our theorem can also help us to determine all extremal graphs for wheels on odd number of vertices. We also discuss the relationship between our results and some previous problems and theorems in spectral graph theory and generalized Turán problems.

本文研究了Bondy的一个著名定理的稳定性结果。证明了对于任意2连通非哈密顿图，如果除最多一个顶点以外的每个顶点度数都至少为k，则除了某些特殊的图族外，它包含一个长度至少为2k+2的循环。我们的结果暗示了几个以前的经典定理，包括沃斯的一个深刻而古老的结果。我们指出Bondy定理稳定性的结果可以直接暗示以下问题的正解（以稍微强一点的形式）：是否存在多项式时间算法来确定n个顶点上的2连通图G是否具有长度至少为min （2δ(G)+2,n}）的循环？这个问题最初激发了Fomin、Golovach、Sagunov和Simonov最近对Dirac定理算法方面的研究，尽管他们用完全不同的方法解决了一个更强大的问题。我们的定理还可以帮助我们确定奇数顶点上的车轮的所有极值图。我们还讨论了我们的结果与谱图理论中的一些问题和定理以及广义Turán问题之间的关系。

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引用次数: 0

Approximate packing of independent transversals in locally sparse graphs 局部稀疏图中独立截线的近似填充

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-07-31 DOI: 10.1016/j.jctb.2025.07.005

Debsoumya Chakraborti , Tuan Tran

Fix

ε > 0

and consider a multipartite graph G with maximum degree at most

(1 - ε) n

, parts

V_{1}, \dots, V_{k}

of the same size n, and where every vertex has at most

o (n)

neighbors in any part

V_{i}

. Loh and Sudakov proved that any such G has an independent transversal. They further conjectured that the vertex set of G can be decomposed into pairwise disjoint independent transversals. In the present paper, we resolve this conjecture approximately by showing that G contains

(1 - ε) n

pairwise disjoint independent transversals. As applications, we give approximate answers to questions of Yuster, and of Fischer, Kühn, and Osthus.

固定ε>；0，考虑一个最大度数为（1−ε）n的多部图G，部分V1，…，Vk的大小为相同n，其中每个顶点在任何部分Vi中最多有o(n)个邻居。Loh和Sudakov证明了任何这样的G都有独立的截线。他们进一步推测，G的顶点集可以分解成两两不相交的独立截线。在本文中，我们通过证明G包含（1−ε）n对不相交的独立截线近似地解决了这个猜想。作为应用，我们给出了Yuster、Fischer、k hn和Osthus问题的近似答案。

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引用次数: 0

Degree-truncated choosability of graphs 图的度截断选择能力

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-07-18 DOI: 10.1016/j.jctb.2025.07.003

Huan Zhou, Jialu Zhu, Xuding Zhu

A graph G is called degree-truncated k-choosable if for every list assignment L with

| L (v) | \geq \min {d_{G} (v), k}

for each vertex v, G is L-colourable. Richter asked whether every 3-connected non-complete planar graph is degree-truncated 6-choosable. We answer this question in negative by constructing a 3-connected non-complete planar graph which is not degree-truncated 7-choosable. Then we prove that every 3-connected non-complete planar graph is degree-truncated 16-DP-colourable (and hence degree-truncated 16-choosable). We further prove that for an arbitrary proper minor closed family

G

of graphs, let s be the minimum integer such that

K_{s, t} \notin G

for some t, then there is a constant k such that every s-connected graph

G \in G

other than a GDP tree is degree-truncated DP-k-colourable (and hence degree-truncated k-choosable), where a GDP-tree is a graph whose blocks are complete graphs or cycles. In particular, for any surface Σ, there is a constant k such that every 3-connected non-complete graph embeddable on Σ is degree-truncated DP-k-colourable (and hence degree-truncated k-choosable). The s-connectedness for graphs in

G

(and 3-connectedness for graphs embeddable on Σ) is necessary, as for any positive integer k,

K_{s - 1, k^{s - 1}} \in G

(

K_{2, k^{2}}

is planar) is not degree-truncated k-choosable. Also, non-completeness is a necessary condition, as complete graphs are not degree-choosable.

图G称为度截断k-可选，如果对于每一个列表赋值L，当|L(v)|≥min （dG(v),k）时，对于每个顶点v， G是L-可着色的。Richter问是否每一个3连通的非完全平面图都是度截断6可选的。通过构造一个不截断7度的3连通非完全平面图，否定地回答了这个问题。然后我们证明了每一个3连通的非完全平面图都是截断度的16- dp可着色的（因此截断度的16- dp是可选择的）。我们进一步证明，对于任意一个图的固有小闭科G，设s为最小整数，使得k，t∈G对于某t，则存在一个常数k，使得除GDP树以外的每一个s连通图G∈G都是截断度的dp -k可着色的（因此是截断度的k可选择的），其中GDP树是其块为完全图或环的图。特别地，对于任意曲面Σ，存在一个常数k，使得每个可嵌入Σ上的3连通非完全图都是度截断的dp -k可着色的（因此度截断的k是可选择的）。G中的图的s连通性（以及可嵌入到Σ上的图的3连通性）是必要的，因为对于任何正整数k， Ks−1,Ks−1∈G （K2， K2是平面的）不是度截断的k可选的。此外，非完备性是一个必要条件，因为完全图是不可度选择的。

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引用次数: 0

Rigid partitions: From high connectivity to random graphs 刚性分区：从高连通性到随机图

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-07-18 DOI: 10.1016/j.jctb.2025.07.001

Michael Krivelevich , Alan Lew , Peleg Michaeli

A graph is called d-rigid if there exists a generic embedding of its vertex set into

R^{d}

such that every continuous motion of the vertices that preserves the lengths of all edges actually preserves the distances between all pairs of vertices. The rigidity of a graph is the maximal d such that the graph is d-rigid. We present new sufficient conditions for the d-rigidity of a graph in terms of the existence of “rigid partitions”—partitions of the graph that satisfy certain connectivity properties. This extends previous results by Crapo, Lindemann, and Lew, Nevo, Peled and Raz.

As an application, we present new results on the rigidity of highly-connected graphs, random graphs, random bipartite graphs, pseudorandom graphs, and dense graphs. In particular, we prove that random

C d \log d

-regular graphs are typically d-rigid, demonstrate the existence of a giant d-rigid component in sparse random binomial graphs, and show that the rigidity of relatively sparse random binomial bipartite graphs is roughly the same as that of the complete bipartite graph, which we consider an interesting phenomenon. Furthermore, we show that a graph admitting

(\begin{matrix} d + 1 \\ 2 \end{matrix})

disjoint connected dominating sets is d-rigid. This implies a weak version of the Lovász–Yemini conjecture on the rigidity of highly-connected graphs. We also present an alternative short proof for a recent result by Lew, Nevo, Peled, and Raz, which asserts that the hitting time for d-rigidity in the random graph process typically coincides with the hitting time for minimum degree d.

一个图被称为d刚性，如果它的顶点集在Rd中有一个一般的嵌入，使得顶点的每一个连续运动都保留了所有边的长度，实际上保留了所有顶点对之间的距离。图的刚性是最大的d，使得图是d刚性的。我们根据“刚性分区”的存在性，给出了图的d-刚性的新充分条件，即图的“刚性分区”满足某些连通性。这扩展了Crapo、Lindemann、Lew、Nevo、Peled和Raz之前的研究结果。作为应用，我们给出了关于高连通图、随机图、随机二部图、伪随机图和密集图的刚性的新结果。特别地，我们证明了随机Cdlog （d-正则图）是典型的d-刚性，证明了稀疏随机二项式图中存在一个巨大的d-刚性分量，并证明了相对稀疏随机二项式二部图的刚性与完全二部图的刚性大致相同，我们认为这是一个有趣的现象。进一步证明了一个包含（d+12）个不相交控制集的图是d刚性的。这意味着关于高连通图的刚性的Lovász-Yemini猜想的一个弱版本。我们还为Lew， Nevo， Peled和Raz最近的结果提供了一个替代的简短证明，该结果断言，随机图过程中d-刚性的命中时间通常与最小度d的命中时间一致。

{"title":"Rigid partitions: From high connectivity to random graphs","authors":"Michael Krivelevich , Alan Lew , Peleg Michaeli","doi":"10.1016/j.jctb.2025.07.001","DOIUrl":"10.1016/j.jctb.2025.07.001","url":null,"abstract":"<div><div>A graph is called d-rigid if there exists a generic embedding of its vertex set into <math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math> such that every continuous motion of the vertices that preserves the lengths of all edges actually preserves the distances between all pairs of vertices. The rigidity of a graph is the maximal d such that the graph is d-rigid. We present new sufficient conditions for the d-rigidity of a graph in terms of the existence of “rigid partitions”—partitions of the graph that satisfy certain connectivity properties. This extends previous results by Crapo, Lindemann, and Lew, Nevo, Peled and Raz.</div><div>As an application, we present new results on the rigidity of highly-connected graphs, random graphs, random bipartite graphs, pseudorandom graphs, and dense graphs. In particular, we prove that random <math><mi>C</mi><mi>d</mi><mi>log</mi><mo>⁡</mo><mi>d</mi></math>-regular graphs are typically d-rigid, demonstrate the existence of a giant d-rigid component in sparse random binomial graphs, and show that the rigidity of relatively sparse random binomial bipartite graphs is roughly the same as that of the complete bipartite graph, which we consider an interesting phenomenon. Furthermore, we show that a graph admitting <math><mo>(</mo><mtable><mtr><mtd><mrow><mi>d</mi><mo>+</mo><mn>1</mn></mrow></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr></mtable><mo>)</mo></math> disjoint connected dominating sets is d-rigid. This implies a weak version of the Lovász–Yemini conjecture on the rigidity of highly-connected graphs. We also present an alternative short proof for a recent result by Lew, Nevo, Peled, and Raz, which asserts that the hitting time for d-rigidity in the random graph process typically coincides with the hitting time for minimum degree d.</div></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"175 ","pages":"Pages 126-170"},"PeriodicalIF":1.2,"publicationDate":"2025-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144654392","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

Fast algorithms for Vizing's theorem on bounded degree graphs 有界度图上Vizing定理的快速算法

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-07-17 DOI: 10.1016/j.jctb.2025.07.002

Anton Bernshteyn , Abhishek Dhawan

Vizing's theorem states that every graph G of maximum degree Δ can be properly edge-colored using

Δ + 1

colors. The fastest currently known

(Δ + 1)

-edge-coloring algorithm for general graphs is due to Sinnamon and runs in time

O (m \sqrt{n})

, where

n ≔ | V (G) |

and

m ≔ | E (G) |

. We investigate the case when Δ is constant, i.e.,

Δ = O (1)

. In this regime, the runtime of Sinnamon's algorithm is

O (n^{3 / 2})

, which can be improved to

O (n \log n)

, as shown by Gabow, Nishizeki, Kariv, Leven, and Terada. Here we give an algorithm whose running time is only

O (n)

, which is obviously best possible. Prior to this work, no linear-time

(Δ + 1)

-edge-coloring algorithm was known for any

Δ ⩾ 4

. Using some of the same ideas, we also develop new algorithms for

(Δ + 1)

-edge-coloring in the

LOCAL

model of distributed computation. Namely, when Δ is constant, we design a deterministic

LOCAL

algorithm with running time

\tilde{O} (\log^{5} n)

and a randomized

LOCAL

algorithm with running time

O (\log^{2} n)

. Although our focus is on the constant Δ regime, our results remain interesting for Δ up to

\log^{o (1)} n

, since the dependence of their running time on Δ is polynomial. The key new ingredient in our algorithms is a novel application of the entropy compression method.

Vizing定理指出，每个最大次为Δ的图G都可以使用Δ+1种颜色来适当地边缘着色。目前已知的最快（Δ+1）的一般图边着色算法是由Sinnamon提出的，运行时间为O(mn)，其中n是对象是|V(G)|， m是对象是|E(G)|。我们研究了Δ为常数的情况，即Δ=O(1)。在这种情况下，Sinnamon算法的运行时间为O(n3/2)，可以改进为O(nlog ln n)，如Gabow、Nishizeki、Kariv、Leven和Terada所示。这里我们给出一个算法，它的运行时间只有O(n)，这显然是最好的可能。在这项工作之前，对于任何Δ大于或等于4的人来说，没有已知的线性时间（Δ+1）边缘着色算法。利用一些相同的思想，我们还在分布式计算的LOCAL模型中开发了（Δ+1）-边缘着色的新算法。即，当Δ为常数时，我们设计了一个运行时间为O ~ （log5 ln）的确定性LOCAL算法和一个运行时间为O（log2 ln）的随机LOCAL算法。虽然我们关注的是常数Δ状态，但我们的结果对于Δ到logo(1) n来说仍然很有趣，因为它们的运行时间对Δ的依赖是多项式的。我们的算法的关键新成分是熵压缩方法的新应用。

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引用次数: 0

Connectoids I: A universal end space theory 连通线I：一个普适的端空间理论

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-06-27 DOI: 10.1016/j.jctb.2025.06.003

Nathan Bowler, Florian Reich

In this series we introduce and investigate the concept of connectoids, which captures the connectivity structure of various discrete objects like undirected graphs, directed graphs, bidirected graphs, hypergraphs or finitary matroids.

In this paper we develop a universal end space theory based on connectoids: the end spaces of connectoids unify the existing end spaces of undirected and directed graphs and establish end spaces for bidirected graphs, hypergraphs and finitary matroids.

The main result shows that the tangle-like description of ends in undirected graphs, called directions, extends to connectoids: there is a one-to-one correspondence between the “directions” of a connectoid and its ends. Furthermore, we generalise normal trees of undirected graphs to connectoids and show that normal trees represent the ends of a connectoid as they do for undirected graphs.

在本系列中，我们介绍并研究了连通图的概念，它捕获了各种离散对象的连通性结构，如无向图、有向图、双向图、超图或有限拟阵。

引用次数: 0

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