Journal of Combinatorial Theory Series B最新文献

英文中文

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 : 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

An infinite family of simple graphs underlying chiral, orientable reflexible and non-orientable rotary maps 手性、可定向、自旋和不可定向旋转映射下的无限简单图族

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-10-16 DOI: 10.1016/j.jctb.2025.10.001

Isabel Hubard , Primož Potočnik , Primož Šparl

In this paper, we provide the first known infinite family of simple graphs, each of which is the skeleton of a chiral map, a skeleton of a reflexible map on an orientable surface, as well as a skeleton of a reflexible map on a non-orientable surface. This family consists of all lexicographic products

C_{n} [m K_{1}]

, where

m \geq 3

n = s m

, with s an integer not divisible by 4. This answers a question posed by Wilson in 2002.

在本文中，我们提供了已知的第一个无限简单图族，每个简单图族都是手性映射的骨架，可定向表面上的自反射映射的骨架，以及不可定向表面上的自反射映射的骨架。这个族由所有字典积Cn[mK1]组成，其中m≥3，n=sm，其中s是不能被4整除的整数。这回答了威尔逊在2002年提出的一个问题。

引用次数: 0

Splitting-off in hypergraphs 超图中的分裂

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-10-09 DOI: 10.1016/j.jctb.2025.09.004

Kristóf Bérczi , Karthekeyan Chandrasekaran , Tamás Király , Shubhang Kulkarni

The splitting-off operation in undirected graphs is a fundamental reduction operation that detaches all edges incident to a given vertex and adds new edges between the neighbors of that vertex while preserving their degrees. Lovász [47], [49] and Mader [50] showed the existence of this operation while preserving global and local connectivities respectively in graphs under certain conditions. These results have far-reaching applications in graph algorithms literature [3], [9], [10], [14], [19], [24], [25], [26], [27], [28], [31], [32], [34], [35], [37], [40], [42], [43], [48], [50], [51], [52], [53]. In this work, we introduce a splitting-off operation in hypergraphs. We show that there exists a local connectivity preserving complete splitting-off in hypergraphs and give a strongly polynomial-time algorithm to compute it in weighted hypergraphs. We illustrate the usefulness of our splitting-off operation in hypergraphs by showing two applications: (1) we give a constructive characterization of k-hyperedge-connected hypergraphs and (2) we give an alternate proof of an approximate min-max relation for max Steiner rooted-connected orientation of graphs and hypergraphs (due to Király and Lau (2008) [40]). Our proof of the approximate min-max relation for graphs circumvents the Nash-Williams' strong orientation theorem and uses tools developed for hypergraphs.

无向图的分离操作是一种基本的约简操作，它将与给定顶点相关的所有边分离，并在该顶点的相邻边之间添加新边，同时保留其度数。Lovász[47]，[49]和Mader[50]分别证明了在一定条件下保持图的全局连通性和局部连通性的情况下，该操作的存在性。这些结果有影响深远的应用图算法文献[3],[9],[10],[14],[19],[24],[25],[26],[27],[28],[31],[32],[34],[35],[37],[40],[42],[43],[48],[50],[51],[52],[53]。在这项工作中，我们在超图中引入了一个分离操作。我们证明了超图中存在一个局部连通性保持完全分离，并给出了一个在加权超图中计算它的强多项式时间算法。我们通过展示两个应用来说明我们的分离操作在超图中的有用性：(1)我们给出了k-超边连接超图的建设性表征；(2)我们给出了图和超图的最大Steiner根连接方向的近似最小-最大关系的替代证明（由于Király和Lau(2008)[40]）。我们对图的近似最小-最大关系的证明绕过了纳什-威廉姆斯的强定向定理，并使用了为超图开发的工具。

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引用次数: 0

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 : 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 : 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 : 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

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 : 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

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 : 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

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 : 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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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 : 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

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