Journal of Combinatorial Theory Series B最新文献

英文中文

Induced subgraphs and tree decompositions XVI. Complete bipartite induced minors 诱导子图和树分解。完全双侧诱导未成年人

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2026-01-01 Epub Date: 2025-10-08 DOI: 10.1016/j.jctb.2025.09.005

Maria Chudnovsky , Sepehr Hajebi , Sophie Spirkl

We prove that for every graph G with a sufficiently large complete bipartite induced minor, either G has an induced minor isomorphic to a large wall, or G contains a large constellation; that is, a complete bipartite induced minor model such that on one side of the bipartition, each branch set is a singleton, and on the other side, each branch set induces a path.

We further refine this theorem by characterizing the unavoidable induced subgraphs of large constellations as two types of highly structured constellations. These results will be key ingredients in several forthcoming papers of this series.

我们证明了对于每一个具有足够大的完全二部诱导小图G，要么G有一个诱导小图同构于一个大墙，要么G包含一个大星座；即，一个完全的二部诱导次要模型，在二分的一侧，每个分支集都是单例，在另一侧，每个分支集都诱导出一条路径。我们进一步完善了这一定理，将大星座的不可避免诱导子图描述为两种高度结构化的星座。这些结果将成为本系列即将发表的几篇论文的关键成分。

引用次数: 0

Triangle Ramsey numbers of complete graphs 完全图的三角形拉姆齐数

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2026-01-01 Epub Date: 2025-10-08 DOI: 10.1016/j.jctb.2025.08.004

Jacob Fox , Jonathan Tidor , Shengtong Zhang

A graph is H-Ramsey if every two-coloring of its edges contains a monochromatic copy of H. Define the F-Ramsey number of H, denoted by

r_{F} (H)

, to be the minimum number of copies of F in a graph which is H-Ramsey. This generalizes the Ramsey number and size Ramsey number of a graph. Addressing a question of Spiro, we prove that

r_{K_{3}} (K_{t}) = (\begin{matrix} r (K_{t}) \\ 3 \end{matrix})

for all sufficiently large t. We do so through a result on graph coloring: there exists an absolute constant K such that every r-chromatic graph where every edge is contained in at least K triangles must contain at least

(\begin{matrix} r \\ 3 \end{matrix})

triangles in total.

定义H的F- ramsey数，用rF(H)表示为H- ramsey图中F的最小拷贝数。这概括了图的Ramsey数和Ramsey数的大小。为了解决Spiro问题，我们证明了对于所有足够大的t， rk3 (Kt)=(r(Kt)3)。我们通过图着色的一个结果证明了这一点：存在一个绝对常数K，使得每个r色图的每条边至少包含K个三角形，必须包含至少（r3）个三角形。

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引用次数: 0

The Hamilton space of pseudorandom graphs 伪随机图的Hamilton空间

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2026-01-01 Epub Date: 2025-09-26 DOI: 10.1016/j.jctb.2025.09.002

Micha Christoph , Rajko Nenadov , Kalina Petrova

We show that if n is odd and

p \geq C \log n / n

, then with high probability Hamilton cycles in

G (n, p)

span its cycle space. More generally, we show this holds for a class of graphs satisfying certain natural pseudorandom properties. The proof is based on a novel idea of parity-switchers, which can be thought of as analogues of absorbers in the context of cycle spaces. As another application of our method, we show that Hamilton cycles in a near-Dirac graph G, that is, a graph G with odd n vertices and minimum degree

n / 2 + C

for sufficiently large constant C, span its cycle space.

我们证明了如果n是奇数且p≥Clog (n, n) /n，那么在G（n,p）中有高概率地张成它的Hamilton环空间。更一般地说，我们证明这适用于一类满足某些自然伪随机性质的图。这个证明是基于奇偶切换器的一个新思想，它可以被认为是循环空间中吸收器的类似物。作为我们方法的另一个应用，我们证明了一个近狄拉克图G中的Hamilton环，即对于足够大的常数C，具有奇数个顶点且最小度为n/2+C的图G，可以张成它的环空间。

引用次数: 0

A proof of a conjecture of Erdős and Gyárfás on monochromatic path covers 关于单色路径覆盖的Erdős和Gyárfás猜想的证明

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2026-01-01 Epub Date: 2025-10-29 DOI: 10.1016/j.jctb.2025.10.007

Alexey Pokrovskiy, Leo Versteegen, Ella Williams

In 1995, Erdős and Gyárfás proved that in every 2-edge-coloured complete graph on n vertices, there exists a collection of

2 \sqrt{n}

monochromatic paths, all of the same colour, which cover the entire vertex set. They conjectured that it is possible to replace

2 \sqrt{n}

\sqrt{n}

. We prove this to be true for all sufficiently large n.

1995年Erdős和Gyárfás证明了在每一个n个顶点的2边彩色完全图中，存在覆盖整个顶点集的2n条颜色相同的单色路径的集合。他们推测用n代替2n是可能的。我们证明对于所有足够大的n都是成立的。

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 : 2025-11-01 Epub 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 : 2025-11-01 Epub 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

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 : 2025-11-01 Epub 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

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 : 2025-11-01 Epub 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。

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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 : 2025-11-01 Epub 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

Hyperbolicity theorems for correspondence colouring 对应着色的双曲性定理

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-11-01 Epub Date: 2025-06-20 DOI: 10.1016/j.jctb.2025.06.002

Luke Postle , Evelyne Smith-Roberge

We generalize a framework of list colouring results to correspondence colouring. Correspondence colouring is a generalization of list colouring wherein we localize the meaning of the colours available to each vertex. As pointed out by Dvořák and Postle, both of Thomassen's theorems on the 5-choosability of planar graphs and 3-choosability of planar graphs of girth at least five carry over to the correspondence colouring setting. In this paper, we show that the family of graphs that are critical for 5-correspondence colouring as well as the family of graphs of girth at least five that are critical for 3-correspondence colouring form hyperbolic families. Analogous results for list colouring were shown by Postle and Thomas and by Dvořák and Kawarabayashi, respectively. Using results on hyperbolic families due to Postle and Thomas, we show that this implies that there exists a universal constant c such that if Σ is a surface of Euler genus g, every graph of edge-width at least

c \cdot \log (g + 1)

embedded in Σ is 5-correspondence colourable. This is asymptotically best possible, and improves upon the best known edge-width bound due to Kim, Kostochka, Li, and Zhu. Using results of Dvořák and Kawarabayashi, we show further that there exist linear-time algorithms for the decidability of 5-correspondence colouring for embedded graphs. We show analogous results for 3-correspondence colouring graphs of girth at least five.

我们将列表着色结果的框架推广到对应着色。对应着色是列表着色的一种推广，其中我们对每个顶点可用的颜色的含义进行局部化。Dvořák和Postle指出，托马森关于平面图形的5-可选性定理和周长至少为5的平面图形的3-可选性定理都适用于对应着色设置。本文证明了5对应着色的临界图族和周长至少为5的3对应着色的临界图族构成双曲族。Postle和Thomas以及Dvořák和Kawarabayashi分别给出了列表着色的类似结果。利用Postle和Thomas关于双曲族的结果，我们证明了这意味着存在一个普适常数c，使得如果Σ是欧拉属g的曲面，则嵌入Σ中的每个边宽至少为c⋅log （g+1）的图都是5对应可着色的。这是渐近最佳可能，并且改进了Kim、Kostochka、Li和Zhu给出的最著名的边宽界。利用Dvořák和Kawarabayashi的结果，我们进一步证明了嵌入图的5对应着色的可判定性存在线性时间算法。我们给出了周长至少为5的3对应着色图的类似结果。

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