Journal of Combinatorial Theory Series B最新文献

英文中文

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

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 : 2025-06-20 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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引用次数: 0

Non-degenerate hypergraphs with exponentially many extremal constructions 具有指数多极值结构的非退化超图

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-06-13 DOI: 10.1016/j.jctb.2025.06.001

József Balogh , Felix Christian Clemen , Haoran Luo

For every integer

t ⩾ 0

, denote by

F_{5}^{t}

the hypergraph on vertex set

{1, 2, \dots, 5 + t}

with hyperedges

{123, 124} \cup {34 k : 5 ⩽ k ⩽ 5 + t}

. We determine

ex (n, F_{5}^{t})

for every

t ⩾ 0

and sufficiently large n and characterize the extremal

F_{5}^{t}

-free hypergraphs. In particular, if n satisfies certain divisibility conditions, then the extremal

F_{5}^{t}

-free hypergraphs are exactly the balanced complete tripartite hypergraphs with additional hyperedges inside each of the three parts

(V_{1}, V_{2}, V_{3})

in the partition; each part

V_{i}

spans a

(| V_{i} |, 3, 2, t)

-design. This generalizes earlier work of Frankl and Füredi on the Turán number of

F_{5} : = F_{5}^{0}

Our results extend a theory of Erdős and Simonovits about the extremal constructions for certain fixed graphs. In particular, the hypergraphs

F_{5}^{6 t}

, for

t ⩾ 1

, are the first examples of hypergraphs with exponentially many extremal constructions and positive Turán density.

对于每个整数t小于0，用F5t表示顶点集{1,2，…，5+t}上的超图，超边{123,124}∪{34k:5≤k≤5+t}。我们为每个t小于0和足够大的n确定ex(n,F5t)，并表征极端无F5t超图。特别地，如果n满足一定的可除性条件，则无f5t极值超图就是分区中三个部分（V1,V2,V3）内各有附加超边的平衡完全三部超图；每个部分Vi跨越一个(|Vi|,3,2,t)-设计。这概括了Frankl和f redi关于F5:=F50的Turán数的早期工作。我们的结果推广了Erdős和Simonovits关于某些固定图的极值结构的理论。特别是，对于t大于或等于1的超图F56t，是具有指数级许多极值结构和正Turán密度的超图的第一个例子。

引用次数: 0

On the Keevash-Knox-Mycroft conjecture 基瓦什-诺克斯-麦考夫猜想

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-05-28 DOI: 10.1016/j.jctb.2025.05.003

Luyining Gan , Jie Han

Given

1 \leq ℓ < k

and

δ \geq 0

, let

PM (k, ℓ, δ)

be the decision problem for the existence of perfect matchings in n-vertex k-uniform hypergraphs with minimum ℓ-degree at least

δ (\begin{matrix} n - ℓ \\ k - ℓ \end{matrix})

. For

k \geq 3

PM (k, ℓ, 0)

was one of the first NP-complete problems by Karp. Keevash, Knox and Mycroft conjectured that

PM (k, ℓ, δ)

is in P for every

δ > 1 - {(1 - 1 / k)}^{k - ℓ}

and verified the case

ℓ = k - 1

In this paper we show that this problem can be reduced to the study of the minimum ℓ-degree condition forcing the existence of fractional perfect matchings. Together with existing results on fractional perfect matchings, this solves the conjecture of Keevash, Knox and Mycroft for

ℓ \geq 0.4 k

. Moreover, we also supply an algorithm that outputs a perfect matching, provided that one exists.

给定1≤l <；k,δ≥0，设PM（k, l,δ）为最小n-度且至少δ(n−k−l)的n顶点k-一致超图中存在完美匹配的判定问题。当k≥3时，PM（k, r,0）是Karp的第一个np完全问题。Keevash， Knox和Mycroft推测，对于每一个δ>；1−(1−1/k)k−r， PM（k, r, r）都在P中，并验证了r =k−1的情况。在本文中，我们证明了这个问题可以简化为对迫使分数阶完美匹配存在的最小阶条件的研究。结合已有的分数阶完美匹配结果，解决了Keevash， Knox和Mycroft对于r≥0.4k的猜想。此外，我们还提供了一个算法，如果存在完美匹配，则输出完美匹配。

引用次数: 0

The 1-2 conjecture holds for regular graphs 1-2猜想适用于正则图

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-05-19 DOI: 10.1016/j.jctb.2025.05.002

Kecai Deng , Hongyuan Qiu

The 1-2 conjecture asserts that the vertices and edges of every graph can be assigned with weights in

{1, 2}

such that adjacent vertices receive distinct weighted degrees. While this conjecture remains open in general, it has been proven that it is possible to achieve this using the weight set

{1, 2, 3}

. We demonstrate that the weight set

{0, 1}

suffices for every graph. As a corollary, the 1-2 conjecture is confirmed for regular graphs. Additionally, we verify another related conjecture concerning locally irregular total colouring, for regular graphs.

1-2猜想断言每个图的顶点和边都可以在{1,2}中赋予权重，使得相邻的顶点得到不同的加权度。虽然这个猜想在一般情况下仍然是开放的，但已经证明可以使用权值集{1,2,3}来实现这一点。我们证明了权值集{0,1}对每个图都是足够的。作为一个推论，对正则图证实了1-2猜想。此外，对于正则图，我们验证了另一个关于局部不规则全着色的相关猜想。

引用次数: 0

Connectivity keeping paths containing prescribed vertices in highly connected triangle-free graphs 在高度连通的无三角形图中保持包含规定顶点的路径的连通性

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-05-16 DOI: 10.1016/j.jctb.2025.05.001

Shinya Fujita

Let

m, k

be integers with

m \geq 1, k \geq 2

. For a k-connected graph G, a subgraph H of G is k-removable if

G - V (H)

is still a k-connected graph. A graph is triangle-free if it contains no triangle as a subgraph.

In this paper, we prove that if G is a k-connected triangle-free graph with minimum degree at least

k + (m - 1) / 2

, then for any vertex

v \in V (G)

, there exists a path P on m vertices starting from v such that

G - V (P)

is a

(k - 1)

-connected graph. This result is obtained by showing a stronger statement concerning the existence of k-removable paths in k-connected triangle-free graphs. We also prove that if G is a k-connected triangle-free graph with minimum degree at least

k + 1

, then G contains a k-removable edge. Our results confirm a conjecture due to Luo et al. concerning the existence of a k-removable path on m vertices in a k-connected bipartite graph for all odd m together with the case

m = 2

设m，k为m≥1，k≥2的整数。对于k连通图G，如果G−V(H)仍然是k连通图，则G的子图H是k可移动的。如果一个图的子图中不包含三角形，那么这个图就是无三角形的。证明了如果G是一个最小度至少为k+(m−1)/2的k连通无三角形图，那么对于任意顶点v∈v (G)，在从v出发的m个顶点上存在一条路径P，使得G−v (P)是一个（k−1）连通图。这一结果是通过证明k连通无三角形图中k条可移动路径的存在性而得到的。我们还证明了如果G是一个k连通且最小度至少为k+1的无三角形图，则G包含一条k可移动边。我们的结果证实了Luo等人关于k连通二部图的m个顶点上存在k个可移动路径的猜想，对于所有奇数m以及m=2的情况。

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引用次数: 0

A structure theorem for pseudosegments and its applications 伪段的一个结构定理及其应用

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-05-06 DOI: 10.1016/j.jctb.2025.04.007

Jacob Fox , János Pach , Andrew Suk

We prove a far-reaching strengthening of Szemerédi's regularity lemma for intersection graphs of pseudosegments. It shows that the vertex set of such a graph can be partitioned into a bounded number of parts of roughly the same size such that almost all bipartite graphs between different pairs of parts are complete or empty. We use this to get an improved bound on disjoint edges in simple topological graphs, showing that every n-vertex simple topological graph with no k pairwise disjoint edges has at most

n {(\log n)}^{O (\log k)}

edges.

我们证明了伪段相交图的szemersamudi正则引理的一个意义深远的强化。证明了这种图的顶点集可以被分割成有限个大小大致相同的部分，使得不同部分对之间的二部图几乎都是完全的或空的。我们用它得到了简单拓扑图中不相交边的改进界，证明了每一个没有k对不相交边的n顶点简单拓扑图最多有n(log log n)O（log k）条边。

引用次数: 0

A characterization of testable hypergraph properties 可测试超图性质的表征

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-05-06 DOI: 10.1016/j.jctb.2025.04.009

Felix Joos , Jaehoon Kim , Daniela Kühn , Deryk Osthus

We provide a combinatorial characterization of all testable properties of k-uniform hypergraphs (k-graphs for short). Here, a k-graph property P is testable if there is a randomized algorithm which makes a bounded number of edge queries and distinguishes with probability 2/3 between k-graphs that satisfy P and those that are far from satisfying P. For the 2-graph case, such a combinatorial characterization was obtained by Alon, Fischer, Newman and Shapira. Our results for the k-graph setting are in contrast to those of Austin and Tao, who showed that for the somewhat stronger concept of local repairability, the testability results for graphs do not extend to the 3-graph setting.

我们提供了k-一致超图（简称k图）的所有可测试性质的组合表征。这里，如果存在一种随机算法，该算法进行有限数量的边查询，并以2/3的概率区分满足P的k图和远远不满足P的k图，则k图性质P是可测试的。对于2图情况，Alon, Fischer， Newman和Shapira获得了这样的组合表征。我们对k图设置的结果与Austin和Tao的结果相反，他们表明，对于更强的局部可修复性概念，图的可测试性结果不能扩展到3图设置。

引用次数: 0

Finding irregular subgraphs via local adjustments 通过局部调整查找不规则子图

IF 1.2 1区数学 Q1 MATHEMATICS

Journal of Combinatorial Theory Series B

Pub Date : 2025-05-02 DOI: 10.1016/j.jctb.2025.04.008

Jie Ma , Shengjie Xie

For a graph H, let

m (H, k)

denote the number of vertices of degree k in H. A conjecture of Alon and Wei states that for any

d \geq 3

, every n-vertex d-regular graph contains a spanning subgraph H satisfying

| m (H, k) - \frac{n}{d + 1} | \leq 2

for every

0 \leq k \leq d

. This holds easily when

d \leq 2

. An asymptotic version of this conjecture was initially established by Frieze, Gould, Karoński and Pfender, subsequently improved by Alon and Wei, and most recently enhanced by Fox, Luo and Pham, approaching its complete range. All of these approaches relied on probabilistic methods.

In this paper, we provide a novel framework to study this conjecture, based on localized deterministic techniques which we call local adjustments. We prove two main results. Firstly, we show that every n-vertex d-regular graph contains a spanning subgraph H satisfying

| m (H, k) - \frac{n}{d + 1} | \leq 2 d^{2}

for all

0 \leq k \leq d

, which provides the first bound independent of the value of n. Secondly, we confirm the case

d = 3

of the Alon-Wei Conjecture in a strong form. Both results can be generalized to multigraphs and yield efficient algorithms for finding the desired subgraphs H. Furthermore, we explore a generalization of the Alon-Wei Conjecture for multigraphs and its connection to the Faudree-Lehel Conjecture concerning irregularity strength.

对于图H，设m（H,k）表示H中k度顶点的个数。Alon和Wei的一个猜想指出，对于任意d≥3，每个n顶点的d正则图都包含一个生成子图H，满足|m(H,k)−和+1|≤2，对于每0≤k≤d。当d≤2时，这很容易成立。这个猜想的渐近版本最初由Frieze， Gould， Karoński和Pfender建立，随后由Alon和Wei改进，最近由Fox， Luo和Pham加强，接近其完整范围。所有这些方法都依赖于概率方法。在本文中，我们提供了一个新的框架来研究这一猜想，基于局部确定性技术，我们称之为局部调整。我们证明了两个主要结果。首先，我们证明了每个n顶点d正则图都包含一个生成子图H满足|m(H,k)−和+1|≤2d2，这提供了与n值无关的第一界。其次，我们以强形式证实了Alon-Wei猜想d=3的情况。这两个结果都可以推广到多图中，并给出了寻找所需子图h的有效算法。此外，我们探讨了多图的Alon-Wei猜想的推广及其与关于不规则强度的Faudree-Lehel猜想的联系。

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