Journal of Combinatorial Theory Series B最新文献

英文中文

When recursion is better than iteration: A linear-time algorithm for directed acyclicity with few error vertices 当递归优于迭代时：一种具有少量错误顶点的有向非循环线性时间算法

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-11-17 DOI: 10.1016/j.jctb.2025.11.002

Daniel Lokshtanov , M.S. Ramanujan , Saket Saurabh

Planarity, bipartiteness and (directed) acyclicity are basic graph properties with classic linear-time recognition algorithms. However, the problems of testing whether a given graph has k vertices whose deletion makes it planar, bipartite or a directed acyclic graph (DAG) are all fundamental NP-complete problems when k is part of the input. As a result, a significant amount of research has been devoted to understanding whether, for every fixed k, these problems admit a polynomial-time algorithm (where the exponent in the polynomial is independent of k) and in particular, whether they admit linear-time algorithms.

While we now know that for every fixed k, we can test in linear time whether a graph is k vertices away from being planar or bipartite, the best known algorithms in the case of directed acyclicity are the algorithm of Garey and Tarjan [IPL 1978], which runs in time

O (n^{k - 1} \cdot m)

and the algorithm of Chen, Liu, Lu, O'Sullivan and Razgon [JACM 2008], which runs in time

O (k! \cdot 4^{k} \cdot k^{4} \cdot n \cdot m)

, where n and m are the number of vertices and arcs in the input digraph, respectively. In other words, it has remained open whether it is possible to recognize in linear time, a graph that is two vertices away from being acyclic.

In this paper, we settle this question by giving an algorithm that decides whether a given graph is k vertices away from being acyclic, in time

O (k! \cdot 4^{k} \cdot k^{5} \cdot (n + m))

. That is, for every fixed k, our algorithm runs in time

O (m + n)

, thus mirroring the case for planarity and bipartiteness.

We obtain our algorithm by introducing a general methodology that shaves off a factor of n from certain algorithms that use the powerful technique of iterative compression. The two main features of our methodology are: (i) This is the first generic technique for designing linear-time FPT algorithms for directed cut problems and (ii) it can be used in combination with future improvements in algorithms for the so-called compression version of other well-studied cut problems such as Multicut and Directed Subset Feedback Vertex Set.

平面性、二分性和（有向）非环性是经典线性时间识别算法的基本图性。然而，当k是输入的一部分时，检验给定图是否有k个顶点的缺失使其成为平面、二部或有向无环图（DAG）的问题都是基本的np完全问题。因此，大量的研究致力于理解，对于每个固定的k，这些问题是否允许多项式时间算法（其中多项式中的指数与k无关），特别是，它们是否允许线性时间算法。

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引用次数: 0

Extremal density for subdivisions with length or sparsity constraints 具有长度或稀疏性约束的细分的极值密度

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-11-17 DOI: 10.1016/j.jctb.2025.11.001

Jaehoon Kim , Hong Liu , Yantao Tang , Guanghui Wang , Donglei Yang , Fan Yang

Given a graph H, a balanced subdivision of H is obtained by replacing all edges of H with internally disjoint paths of the same length. In this paper, we prove that for any graph H, a linear-in-

e (H)

bound on average degree guarantees a balanced H-subdivision. This strengthens an old result of Bollobás and Thomason, and resolves a question of Gil-Fernández, Hyde, Liu, Pikhurko and Wu.

We observe that this linear bound on average degree is best possible whenever H is logarithmically dense. We further show that this logarithmic density is the critical threshold: for many graphs H below this density, its subdivisions are forcible by a sublinear-in-

e (H)

bound on average degree. We provide such examples by proving that the subdivisions of any almost bipartite graph H with sublogarithmic density are forcible by a sublinear-in-

e (H)

bound on average degree, provided that H satisfies some additional separability condition.

给定图H，通过将H的所有边替换为相同长度的内部不相交路径，得到H的均衡细分。本文证明了对于任意图H，在平均度上的线性- In -e(H)界保证了均衡的H细分。这加强了Bollobás和Thomason的一个老结果，解决了Gil-Fernández、Hyde、Liu、Pikhurko和Wu的问题。

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引用次数: 0

Reconfiguration of basis pairs in regular matroids 正则拟阵中基对的重构

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-11-14 DOI: 10.1016/j.jctb.2025.10.009

Kristóf Bérczi , Bence Mátravölgyi , Tamás Schwarcz

In recent years, combinatorial reconfiguration problems have attracted great attention due to their connection to various topics such as optimization, counting, enumeration, or sampling. One of the most intriguing open questions concerns the exchange distance of two matroid basis sequences, a problem that appears in several areas of computer science and mathematics. White (1980) proposed a conjecture for the characterization of two basis sequences being reachable from each other by symmetric exchanges, which received a significant interest also in algebra due to its connection to toric ideals and Gröbner bases. In this work, we verify White's conjecture for basis sequences of length two in regular matroids, a problem that was formulated as a separate question by Farber, Richter, and Shank (1985) and Andres, Hochstättler, and Merkel (2014). Most of previous work on White's conjecture has not considered the question from an algorithmic perspective. We study the problem from an optimization point of view: our proof implies a polynomial algorithm for determining a sequence of symmetric exchanges that transforms a basis pair into another, thus providing the first polynomial upper bound on the exchange distance of basis pairs in regular matroids. As a byproduct, we verify a conjecture of Gabow (1976) on the serial symmetric exchange property of matroids for the regular case.

近年来，组合重构问题由于与优化、计数、枚举或抽样等各种主题的联系而引起了人们的广泛关注。最有趣的开放问题之一涉及两个矩阵基序列的交换距离，这个问题出现在计算机科学和数学的几个领域。White（1980）提出了一个关于两个基序列通过对称交换可相互到达的表征的猜想，由于它与环态理想和Gröbner基的联系，该猜想在代数中也引起了极大的兴趣。在这项工作中，我们验证了White关于正则拟阵中长度为2的基序列的猜想，这个问题被Farber， Richter, and Shank (1985), Andres, Hochstättler, and Merkel（2014）作为一个单独的问题公式化。怀特猜想之前的大部分工作都没有从算法的角度考虑这个问题。我们从最优化的角度研究了这个问题：我们的证明包含了一个多项式算法，用于确定将一个基对转化为另一个基对的对称交换序列，从而提供了正则拟阵中基对交换距离的第一个多项式上界。作为一个副产品，我们在正则情况下验证了Gabow（1976）关于拟阵的序列对称交换性质的一个猜想。

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引用次数: 0

The Maker-Breaker percolation game on the square lattice 方块格子上的Maker-Breaker渗透游戏

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-11-11 DOI: 10.1016/j.jctb.2025.10.010

Vojtěch Dvořák, Adva Mond, Victor Souza

We study the

(m, b)

Maker-Breaker percolation game on

Z^{2}

, introduced by Day and Falgas-Ravry. In this game, on each of their turns, Maker and Breaker claim respectively m and b unclaimed edges of the square lattice

Z^{2}

. Breaker wins if the component containing the origin becomes finite when his edges are deleted from

Z^{2}

. Maker wins if she can indefinitely avoid a win of Breaker. We show that Breaker has a winning strategy for the

(m, b)

game whenever

b ⩾ (2 - 1 / 14 + o (1)) m

, breaking the ratio 2 barrier proved by Day and Falgas-Ravry.

Addressing further questions of Day and Falgas-Ravry, we show that Breaker can win the

(m, 2 m)

game even if he allows Maker to claim c edges before the game starts, for any integer c, and that he can moreover win rather quickly as a function of c.

We also consider the game played on the so-called polluted board, obtained after performing Bernoulli bond percolation on

Z^{2}

with parameter p. We show that for the

(1, 1)

game on the polluted board, Breaker almost surely has a winning strategy whenever

p ⩽ 0.6298

我们研究了Day和Falgas-Ravry在Z2上引入的（m,b） Maker-Breaker渗透博弈。在这个游戏中，制造者和破坏者在各自的回合中分别占有方形格子Z2的m条和b条无人认领的边。如果包含原点的组件在其边缘从Z2中删除时变得有限，则断路器获胜。如果制造者可以无限期地避免毁灭者的胜利，她就赢了。我们表明，每当b大于或等于（2−1/14+o(1)）m时，Breaker具有（m,b）游戏的制胜策略，打破了Day和Falgas-Ravry证明的比率2障碍。为了进一步解决Day和Falgas-Ravry的问题，我们证明了对于任何整数c，即使Breaker允许Maker在游戏开始前声称c条边，他也可以赢得（m,2m）游戏，并且作为c的函数，他还可以相当快地获胜。我们还考虑了在所谓的污染棋盘上进行的游戏，该游戏是在参数p的Z2上执行伯努利键渗透后获得的。我们表明，对于污染棋盘上的（1,1）游戏，当p≤0.6298时，破局者几乎肯定有一个获胜策略。

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引用次数: 0

Cubic graphs with no eigenvalues in the interval (−1,1) 在（−1,1）区间内无特征值的三次图

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-11-05 DOI: 10.1016/j.jctb.2025.10.008

Krystal Guo , Gordon F. Royle

We give a complete characterization of the cubic graphs with no eigenvalues in the open interval

(- 1, 1)

. We first classify the connected cubic graphs with no eigenvalues in

(- 1, 1)

showing that there are two infinite families: one due to Guo and Mohar (2014) [7] and the other due to Kollár and Sarnak (2021) [12], and 13 “sporadic” graphs on at most 32 vertices. Then a not necessarily connected cubic graph has no eigenvalues in

(- 1, 1)

if and only if the same is true for every connected component. This classification allows us to show that

(- 1, 1)

is a maximal spectral gap set for cubic graphs, thereby answering a question of Kollár and Sarnak (2021) [12]. The techniques used include examination of the small subgraphs that can appear in such a graph and an application of the classification of generalized line graphs.

我们给出了开区间（- 1,1）上无特征值的三次图的完整刻划。我们首先对（- 1,1）中没有特征值的连通三次图进行分类，表明存在两个无限族：一个是由于Guo和Mohar(2014)[7]，另一个是由于Kollár和Sarnak(2021)[12]，以及13个“偶发”图，最多32个顶点。则一个不一定连通的三次图在（- 1,1）中没有特征值，当且仅当对每个连通分量都成立。这种分类使我们能够证明（−1,1）是三次图的最大谱间隙集，从而回答了Kollár和Sarnak(2021)[12]的问题。所使用的技术包括检查可能出现在这种图中的小子图和广义线形图分类的应用。

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引用次数: 0

Non-linear Hamilton cycles in linear quasirandom and uniformly dense hypergraphs 线性准随机和均匀稠密超图中的非线性Hamilton环

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-11-05 DOI: 10.1016/j.jctb.2025.10.006

Jie Han , Xichao Shu , Guanghui Wang

A k-graph H is called

(p, μ)

-dense if for all not necessarily distinct sets

A_{1}, \dots, A_{k} \subseteq V (H)

we have

e (A_{1}, \dots, A_{k}) \geq p | A_{1} | \dots | A_{k} | - μ | V (H) |^{k}

. This is believed to be the weakest form of quasirandomness in k-graphs and also known as linear quasirandomness.

In this paper, we show that for

ℓ < k

satisfying

(k - ℓ) ∤ k

(p, μ)

-density plus a minimum

(ℓ + 1)

-degree of

α n^{k - ℓ - 1}

guarantees Hamilton ℓ-cycles, but requiring a minimum ℓ-degree of

Ω (n^{k - ℓ})

instead is not sufficient. This answers a question of Lenz–Mubayi–Mycroft and characterizes the triples

(k, ℓ, d)

when

k - ℓ ∤ k

such that degenerate choices of p and α force ℓ-Hamiltonicity.

We actually prove a general result on ℓ-Hamiltonicity in quasirandom k-graphs, assuming a minimum vertex degree and essentially that every two ℓ-sets can be connected by a constant length ℓ-path. This reduces the ℓ-Hamiltonicity problem to the study of the connection property which also allows us to deduce a

(k / 2)

-Hamiltonicity result in uniformly dense k-graphs (for even

k \geq 4

Our proof uses the lattice-based absorption method in the non-standard way and is the first one that embeds a non-linear Hamilton cycle in linear quasirandom k-graphs.

如果对于所有不一定不同的集合A1，…，Ak，我们有e(A1，…，Ak)≥p|A1|⋯|Ak|−μ|V(H)|k，则称k图H为（p,μ）密集。这被认为是k图中最弱的拟随机形式，也被称为线性拟随机。在本文中，我们证明了对于满足(k−r)∤k, (p,μ)-密度加上αnk−r - 1的最小(r +1)-度保证了Hamilton r -环，但要求最小r -度为Ω（k−r）是不够的。这回答了Lenz-Mubayi-Mycroft的一个问题，并描述了当k−r∤k时三元组（k, r,d）使得p和α力的选择退化为r -哈密顿性。我们实际上证明了一个关于准随机k图中r -哈密性的一般结果，假设有一个最小顶点度，并且本质上每两个r -集合都可以通过一个常数长度的r -路径连接起来。这就将l -哈密性问题简化为对连接性质的研究，这也使我们能够在均匀密集的k图中推导出(k/2)-哈密性结果（即使k≥4）。我们的证明以非标准的方式使用基于格的吸收方法，并且是第一个在线性准随机k图中嵌入非线性汉密尔顿循环的证明。

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 : 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

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 : 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

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 : 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

Asymptotic half-grid and full-grid minors 渐近半格和全格子

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-10-22 DOI: 10.1016/j.jctb.2025.10.003

Sandra Albrechtsen, Matthias Hamann

We prove that every locally finite, quasi-transitive graph with a thick end whose cycle space is generated by cycles of bounded length contains the full-grid as an asymptotic minor and as a diverging minor. This in particular includes all locally finite Cayley graphs of finitely presented groups that are not virtually free, and partially solves problems of Georgakopoulos and Papasoglu and of Georgakopoulos and Hamann.

Additionally, we show that every (not necessarily quasi-transitive) graph of finite maximum degree which has a thick end and whose cycle space is generated by cycles of bounded length contains the half-grid as an asymptotic minor and as a diverging minor.

证明了每一个具有厚端的局部有限拟传递图，其循环空间是由有界长度的循环生成的，其全网格是渐近小格和发散小格。这尤其包括所有有限表示群的局部有限Cayley图，这些群不是几乎自由的，并且部分地解决了Georgakopoulos和Papasoglu以及Georgakopoulos和Hamann的问题。此外，我们还证明了每一个（不一定是拟传递的）有一个厚端且其循环空间由有界长度的循环生成的有限最大度图都包含半网格作为渐近次元和发散次元。

引用次数: 0

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