Journal of Combinatorial Theory Series B最新文献

英文中文

Thomassen's theorem on the two-linkage problem in acyclic digraphs: A shorter proof 无环有向图中双连杆问题的Thomassen定理：一个简短的证明

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2026-01-01 Epub Date: 2025-09-11 DOI: 10.1016/j.jctb.2025.08.006

Paul Seymour

Let G be an acyclic digraph, and let

a, b, c, d \in V (G)

, where

a, b

are sources,

c, d

are sinks, and every other vertex has in-degree and out-degree at least two. In 1985, Thomassen showed that there do not exist disjoint directed paths from a to c and from b to d, if and only if G can be drawn in a closed disc with

a, b, c, d

drawn in the boundary in order. We give a shorter proof.

设G是一个无环有向图，设a，b,c,d∈V(G)，其中a，b为源，c，d为汇，且其他每一个顶点至少有两个入度和出度。1985年，Thomassen证明了不存在从a到c和从b到d的不相交的有向路径，当且仅当G可以画在封闭圆盘上，a,b,c，d依次画在边界上。我们给出一个简短的证明。

引用次数: 0

Transversals via regularity 通过规则的截线

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.2025.10.004

Yangyang Cheng , Katherine Staden

Given graphs

G_{1}, \dots, G_{s}

all on the same vertex set and a graph H with

e (H) \leq s

, a copy of H is transversal or rainbow if it contains at most one edge from each

G_{c}

. We study the case when H is spanning and explore how the regularity blow-up method, that has been so successful in the uncoloured setting, can be used to find transversals. We provide the analogues of the tools required to apply this method in the transversal setting. Our main result is a blow-up lemma for transversals that applies to separable bounded degree graphs H.

Our proofs use weak regularity in the 3-uniform hypergraph whose edges are those xyc where xy is an edge in the graph

G_{c}

. We apply our lemma to give a large class of spanning 3-uniform linear hypergraphs H such that any sufficiently large uniformly dense n-vertex 3-uniform hypergraph with minimum vertex degree

Ω (n^{2})

contains H as a subhypergraph. This extends work of Lenz, Mubayi and Mycroft.

给定图G1，…，Gs都在同一个顶点集上，图H e(H)≤s，如果H的副本最多包含来自每个Gc的一条边，则它是截线或彩虹。我们研究了H生成时的情况，并探索了如何使用正则性放大方法来寻找截线，这种方法在未着色的情况下非常成功。我们提供了在横向设置中应用该方法所需的工具的类似物。我们的主要结果是一个适用于可分离有界度图h的截线的爆破引理。我们的证明使用了3-一致超图的弱正则性，其边是那些xyc，其中xy是图Gc中的一条边。我们应用引理给出了一个大的生成3-一致线性超图H，使得任何足够大的具有最小顶点度Ω（n2）的一致密集n顶点3-一致超图都包含H作为子超图。这扩展了Lenz， Mubayi和Mycroft的工作。

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引用次数: 0

Detachable pairs in 3-connected matroids and simple 3-connected graphs 3连通拟阵和简单3连通图中的可分离对

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2026-01-01 Epub Date: 2025-09-23 DOI: 10.1016/j.jctb.2025.09.003

Nick Brettell , Charles Semple , Gerry Toft

Let M be a 3-connected matroid. A pair

{e, f}

in M is detachable if

M ﹨ e ﹨ f

M / e / f

is 3-connected. Williams (2015) proved that if M has at least 13 elements, then at least one of the following holds: M has a detachable pair, M has a 3-element circuit or cocircuit, or M is a spike. We address the case where M has a 3-element circuit or cocircuit, to obtain a characterisation of when a matroid with at least 13 elements has a detachable pair. As a consequence, we characterise when a simple 3-connected graph G with

| E (G) | \geq 13

has a pair of edges

{e, f}

such that

G / e / f

G ﹨ e ﹨ f

is simple and 3-connected.

设M是一个3连通的矩阵。如果Mef或M/e/f为3连通，则M中的一对{e，f}是可分离的。Williams（2015）证明，如果M至少有13个元件，则M有一个可拆卸的对，M有一个3元电路或共电路，或M是一个尖峰。我们处理M具有3元电路或共电路的情况，以获得具有至少13个单元的矩阵何时具有可拆卸对的特征。因此，我们刻画了当一个|E(G)|≥13的简单3连通图G有一对边{E，f}使得G E f的G/ E /f是简单3连通的。

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引用次数: 0

Quantum advantage and CSP complexity 量子优势和CSP复杂性

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2026-01-01 Epub Date: 2025-10-17 DOI: 10.1016/j.jctb.2025.10.002

Lorenzo Ciardo

Information-processing tasks modelled by homomorphisms between relational structures can witness quantum advantage when entanglement is used as a computational resource. We prove that the occurrence of quantum advantage is determined by the same algebraic structure (known as the polymorphism minion) that captures the complexity of CSPs. We investigate the connection between the minion of quantum advantage and other known minions controlling CSP tractability and width. In this way, we make use of complexity results from the algebraic theory of CSPs to characterise the occurrence of quantum advantage in the case of graphs, and to obtain new necessary and sufficient conditions in the case of arbitrary relational structures.

将纠缠作为一种计算资源，利用关系结构间同态建模的信息处理任务具有量子优势。我们证明了量子优势的发生是由捕获csp复杂性的相同代数结构（称为多态性仆从）决定的。我们研究了量子优势小黄人与控制CSP可牵引性和宽度的其他已知小黄人之间的联系。这样，我们利用csp代数理论的复杂性结果来刻画图的情况下量子优势的发生，并在任意关系结构的情况下得到新的充要条件。

引用次数: 0

Local properties of the spectral radius and Perron vector in graphs 图中谱半径和Perron向量的局部性质

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2026-01-01 Epub Date: 2025-09-25 DOI: 10.1016/j.jctb.2025.09.001

Lele Liu , Bo Ning

In 2002, Nikiforov proved that for an n-vertex graph G with clique number ω and edge number m, its spectral radius

λ (G)

satisfies

λ (G) \leq \sqrt{2 (1 - 1 / ω) m}

, which confirmed a conjecture implicitly suggested by Edwards and Elphick. In this paper, we prove a local version of spectral Turán inequality, showing that

λ^{2} (G) \leq 2 \sum_{e \in E (G)} \frac{c (e) - 1}{c (e)}

, where

c (e)

is the order of the largest clique containing the edge e in G. We also characterize the extremal graphs. Furthermore, we prove that our theorem implies Nikiforov's theorem and provide an example in which the difference of Nikiforov's bound and ours is

Ω (\sqrt{m})

for some cases. Our second result explores local properties of the Perron vector of graphs. We disprove a conjecture of Gregory, asserting that for a connected n-vertex graph G with chromatic number

k \geq 2

and an independent set S, we have

\sum_{v \in S} x_{v}^{2} \leq \frac{1}{2} - \frac{k - 2}{2 \sqrt{{(k - 2)}^{2} + 4 (k - 1) (n - k + 1)}},

where

x_{v}

is the component of the Perron vector of G with respect to the vertex v. A modified version of Gregory's conjecture is proposed.

2002年，Nikiforov证明了对于团数为ω，边数为m的n顶点图G，其谱半径λ(G)满足λ(G)≤2(1−1/ω)m，证实了Edwards和Elphick隐式提出的一个猜想。本文证明了谱Turán不等式的一个局部版本，证明了λ2(G)≤2∑e∈e (G)c(e)−1c(e)，其中c(e)是G中包含边e的最大团的阶，并刻画了极值图。进一步证明了我们的定理蕴涵了Nikiforov定理，并给出了在某些情况下Nikiforov界与我们的界之差为Ω(m)的一个例子。我们的第二个结果探讨了图的Perron向量的局部性质。我们证明了Gregory的一个猜想，证明了对于色数k≥2的连通n顶点图G和独立集S，有∑v∈Sxv2≤12−k−22(k−2)2+4(k−1)(n−k+1)，其中xv是G关于顶点v的Perron向量的分量。

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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 : 2026-01-01 Epub 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

Tree independence number II. Three-path-configurations 树的独立性2。Three-path-configurations

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2026-01-01 Epub Date: 2025-09-08 DOI: 10.1016/j.jctb.2025.08.003

Maria Chudnovsky , Sepehr Hajebi , Daniel Lokshtanov , Sophie Spirkl

A three-path-configuration is a graph consisting of three pairwise internally-disjoint paths the union of every two of which is an induced cycle of length at least four. A graph is 3PC-free if no induced subgraph of it is a three-path-configuration. We prove that 3PC-free graphs have poly-logarithmic tree independence number. More explicitly, we show that there exists a constant c such that every n-vertex 3PC-free graph has a tree decomposition in which every bag has stability number at most

c {(\log n)}^{2}

. This implies that the Maximum Weight Independent Set problem, as well as several other natural algorithmic problems, that are known to be NP-hard in general, can be solved in quasi-polynomial time if the input graph is 3PC-free.

一个三路径构型是一个图，它由三条对的内部不相交的路径组成，其中每两条路径的并集是一个长度至少为4的诱导环。如果图的任何诱导子图都不是三路径配置，则该图是无3pc的。证明了无3pc图具有多对数树独立数。更明确地说，我们证明了存在一个常数c，使得每个无n顶点3pc的图都有一个树分解，其中每个袋的稳定数最多为c（log (n)2）。这意味着，如果输入图是无3pc的，那么最大权重独立集问题，以及其他一些已知一般是np困难的自然算法问题，可以在拟多项式时间内解决。

引用次数: 0

Dense minors of graphs with independence number two 具有独立性为2的图的密集子图

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2026-01-01 Epub Date: 2025-09-17 DOI: 10.1016/j.jctb.2025.08.005

Sergey Norin , Paul Seymour

Motivated by Hadwiger's conjecture, we prove that every graph with no independent set of size three contains a t-vertex simple minor with

0.98688 \cdot (\begin{matrix} t \\ 2 \end{matrix}) - o (t^{2})

edges, where t is its chromatic number.

根据Hadwiger的猜想，我们证明了每一个没有大小为3的独立集的图都包含一个具有0.98688⋅(t2)−o（t2）条边的t顶点简单小图，其中t为其色数。

引用次数: 0

Counterexamples to the linkage conjecture for tournaments 关于比赛的关联猜想的反例

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2026-01-01 Epub Date: 2025-10-27 DOI: 10.1016/j.jctb.2025.10.005

Jia Zhou, Jin Yan

Let

k \geq 2

be an integer. A digraph D is k-linked if for every set of 2k distinct vertices

x_{1}, \dots, x_{k}, y_{1}, \dots, y_{k}

in D, there exist k pairwise vertex-disjoint paths

P_{1}, \dots, P_{k}

such that each path

P_{i}

starts at

x_{i}

and ends at

y_{i}

for

i \in [k]

. In 2015, Pokrovskiy conjectured that there exists a function

g (k)

such that every 2k-connected tournament with minimum in-degree and minimum out-degree at least

g (k)

is k-linked in Pokrovskiy (2015) [16]. In this paper, we disprove Pokrovskiy's conjecture by constructing a family of 2k-connected tournaments of order

n \geq 14 k^{2}

with arbitrarily large minimum semi-degree (depending on n) that are not k-linked. The counterexamples, with sufficiently large order n, also provide a negative answer to the question posed by Girão et al. (2021) [8]: whether or not 2k-connectivity is sufficient for k-linkage in every tournament with minimum out-degree at least some polynomial in k.

设k≥2为整数。有向图D是k链接的，如果对于D中每个由2k个不同的顶点x1，…，xk,y1，…，yk组成的集合，存在k个成对的顶点不相交路径P1，…，Pk，使得对于i∈[k]，每个路径Pi从xi开始，以yi结束。2015年，Pokrovskiy推测存在一个函数g(k)，使得Pokrovskiy(2015)[16]中每一个最小入度和最小出度至少为g(k)的2k连通锦标赛都是k连通的。在本文中，我们通过构造一个非k连接的具有任意大的最小半度（取决于n）的n≥14k2阶的2k连接竞赛族来反驳Pokrovskiy猜想。反例，具有足够大的n阶，也为gir等人（2021）[8]提出的问题提供了一个否定的答案：在k中至少有某个多项式的最小出位度的每个锦标赛中，2k-连通性是否足以满足k-链接。

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引用次数: 0

The degree-restricted random process is far from uniform 受程度限制的随机过程远非一致

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2026-01-01 Epub Date: 2025-09-23 DOI: 10.1016/j.jctb.2025.08.001

Michael Molloy , Erlang Surya , Lutz Warnke

The degree-restricted random process is a natural algorithmic model for generating graphs with degree sequence

d_{n} = (d_{1}, \dots, d_{n})

: starting with an empty n-vertex graph, it sequentially adds new random edges so that the degree of each vertex

v_{i}

remains at most

d_{i}

. Wormald conjectured in 1999 that, for d-regular degree sequences

d_{n}

, the final graph of this process is similar to a uniform random d-regular graph.

In this paper we show that, for degree sequences

d_{n}

that are not nearly regular, the final graph of the degree-restricted random process differs substantially from a uniform random graph with degree sequence

d_{n}

. The combinatorial proof technique is our main conceptual contribution: we adapt the switching method to the degree-restricted process, demonstrating that this enumeration technique can also be used to analyze stochastic processes (rather than just uniform random models, as before).

度限制随机过程是生成度序列dn=（d1，…，dn）图的一种自然算法模型：从一个空的n顶点图开始，顺序地添加新的随机边，使每个顶点vi的度最多保持di。Wormald在1999年推测，对于d正则次序列dn，该过程的最终图类似于一致随机d正则图。在本文中，我们证明了对于不接近正则的次序列dn，限制次随机过程的最终图与具有次序列dn的一致随机图有很大的不同。组合证明技术是我们的主要概念贡献：我们将切换方法适应于程度限制过程，证明这种枚举技术也可以用于分析随机过程（而不仅仅是均匀随机模型，就像以前一样）。

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