European Journal of Combinatorics最新文献

英文中文

Proof of a conjecture on the shape-Wilf-equivalence for partially ordered patterns 部分有序模式的形状-威尔夫等价的一个猜想的证明

IF 1 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2025-12-01 Epub Date: 2025-07-09 DOI: 10.1016/j.ejc.2025.104222

Lintong Wang, Sherry H.F. Yan

A partially ordered pattern (abbreviated POP) is a partially ordered set (poset) that generalizes the notion of a pattern when we are not concerned with the relative order of some of its letters. The notion of partially ordered patterns provides a convenient language to deal with large sets of permutation patterns. In analogy to the shape-Wilf-equivalence for permutation patterns, Burstein–Han–Kitaev–Zhang initiated the study of the shape-Wilf-equivalence for POPs which would result in the shape-Wilf-equivalence for large sets of permutation patterns. The main objective of this paper is to confirm a recent intriguing conjecture posed by Burstein–Han–Kitaev–Zhang concerning the shape-Wilf-equivalence for POPs of length

k

. This is accomplished by establishing a bijection between two sets of pattern-avoiding transversals of a given Young diagram.

部分有序模式（简称POP）是一种部分有序集合（poset），当我们不关心其中一些字母的相对顺序时，它概括了模式的概念。部分有序模式的概念提供了一种方便的语言来处理大量排列模式集。与排列模式的形状-威尔夫等价类似，Burstein-Han-Kitaev-Zhang发起了持久性有机污染物的形状-威尔夫等价研究，这将导致大排列模式集的形状-威尔夫等价。本文的主要目的是证实最近由Burstein-Han-Kitaev-Zhang提出的关于长度为k的pop的形状- wilf等价的有趣猜想。这是通过在给定Young图的两组避模截线之间建立双射来完成的。

引用次数: 0

Stability properties for subgroups generated by return words 由返回字生成的子组的稳定性

IF 1 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2025-12-01 Epub Date: 2025-07-26 DOI: 10.1016/j.ejc.2025.104224

France Gheeraert , Herman Goulet-Ouellet , Julien Leroy , Pierre Stas

Return words are a classical tool for studying shift spaces with low factor complexity. In recent years, their projection inside groups have attracted some attention, for instance in the context of dendric shift spaces, of generation of pseudorandom numbers (through the welldoc property), and of profinite invariants of shift spaces. Aiming at unifying disparate works, we introduce a notion of stability for subgroups generated by return words. Within this framework, we revisit several existing results and generalize some of them. We also study general aspects of stability, such as decidability or closure under certain operations.

返回词是研究低因子复杂度移位空间的经典工具。近年来，它们在群内的投影引起了一些关注，例如在枝状移位空间，伪随机数的生成（通过welldoc性质）以及移位空间的无限不变量的背景下。为了统一不同的作品，我们引入了由返回词生成的子群的稳定性概念。在这个框架内，我们回顾了几个现有的结果，并概括了其中的一些。我们还研究了稳定性的一般方面，例如在某些操作下的可判决性或闭包性。

引用次数: 0

Extensions and applications of the Tuza-Vestergaard theorem tuza - vesterggaard定理的扩展与应用

IF 1 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2025-12-01 Epub Date: 2025-07-01 DOI: 10.1016/j.ejc.2025.104201

Michael A. Henning , Anders Yeo

<div><div>The transversal number <math><mrow><mi>τ</mi><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow></mrow></math> of a hypergraph <math><mi>H</mi></math> is the minimum number of vertices that intersect every edge of <math><mi>H</mi></math>. A 6-uniform hypergraph has all edges of size 6. On 10 November 2000 Tuza and Vestergaard (2002) conjectured that if <math><mi>H</mi></math> is a 3-regular 6-uniform hypergraph of order <math><mi>n</mi></math>, then <math><mrow><mi>τ</mi><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow><mo>≤</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac><mi>n</mi></mrow></math>. This conjecture was recently proven by the Henning and Yeo (2023) and is now called the Tuza-Vestergaard Theorem. In this paper we extend the Tuza-Vestergaard Theorem by relaxing the 3-regularity constraint and allowing bounded maximum degree 4. We present several applications of the Tuza-Vestergaard Theorem and its extension. We obtain best known upper bounds to date on the transversal number of a (general) 6-uniform hypergraph <math><mi>H</mi></math> of order <math><mi>n</mi></math> and size <math><mi>m</mi></math>. In particular, if <math><mi>H</mi></math> is a 4-regular 6-uniform hypergraph of order <math><mi>n</mi></math>, then we show that <math><mrow><mi>τ</mi><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow><mo>≤</mo><mfrac><mrow><mn>2</mn></mrow><mrow><mn>7</mn></mrow></mfrac><mi>n</mi></mrow></math>. The Tuza constant <math><msub><mrow><mi>c</mi></mrow><mrow><mn>6</mn></mrow></msub></math> is defined by <math><mrow><msub><mrow><mi>c</mi></mrow><mrow><mn>6</mn></mrow></msub><mo>=</mo><mo>sup</mo><mfrac><mrow><mi>τ</mi><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow></mrow><mrow><mi>n</mi><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow><mo>+</mo><mi>m</mi><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow></mrow></mfrac></mrow></math>, where the supremum is taken over the class of all 6-uniform hypergraphs <math><mi>H</mi></math>. Since 1990 the exact value of <math><msub><mrow><mi>c</mi></mrow><mrow><mn>6</mn></mrow></msub></math> has yet to be determined. We show that <math><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>6</mn></mrow></mfrac><mo>≤</mo><msub><mrow><mi>c</mi></mrow><mrow><mn>6</mn></mrow></msub><mo>≤</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>6</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>210</mn></mrow></mfrac></mrow></math>, where <math><mrow><msub><mrow><mi>c</mi></mrow><mrow><mn>6</mn></mrow></msub><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>6</mn></mrow></mfrac></mrow></math> is conjectured to be the correct bound. Moreover we show that if <math><mi>G</mi></math> is a graph of order <math><mi>n</mi></math> with <math><mrow><mi>δ</mi><mrow><mo>(<

超图H的截线数τ(H)是与H的每条边相交的最小顶点数。一个6-均匀超图的所有边的长度为6。2000年11月10日，Tuza和vesterggaard（2002）推测，如果H是一个n阶的3-正则6-一致超图，则τ(H)≤14n。这个猜想最近被Henning和Yeo（2023）证明，现在被称为tuza - vesterggaard定理。本文通过放宽3正则约束并允许有界最大次4，扩展了tuza - vesterggaard定理。给出了tuza - vesterggaard定理及其推广的几个应用。我们得到了大小为m的n阶（一般）6-均匀超图H的截数的已知上界。特别地，如果H是n阶的4-正则6-均匀超图，则我们证明了τ(H)≤27n。图萨常数c6定义为c6=supτ(H)n(H)+m(H)，其中，所有6-一致超图H的类都取至极值。自1990年以来，c6的确切值尚未确定。我们证明了16≤c6≤16+1210，其中c6=16被推测为正确的界。进一步证明了如果G是n阶图且δ(G)≥6，则γt(G)≤413+6217n，其中γt(G)表示G的总支配数，推测γt(G)≤413n为正确界。这些边界改进了迄今为止最知名的边界。

引用次数: 0

Faces in rectilinear drawings of complete graphs 完全图的直线图中的面

IF 1 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2025-12-01 Epub Date: 2025-07-11 DOI: 10.1016/j.ejc.2025.104217

Martin Balko , Anna Brötzner , Fabian Klute , Josef Tkadlec

We initiate the study of extremal problems about faces in convex rectilinear drawings of

K_{n}

, that is, drawings where vertices are represented by points in the plane in convex position and edges by line segments between the points representing the end-vertices. We show that if a convex rectilinear drawing of

K_{n}

does not contain a common interior point of at least three edges, then there is always a face forming a convex 5-gon while there are such drawings without any face forming a convex

k

-gon with

k \geq 6

A convex rectilinear drawing of

K_{n}

is regular if its vertices correspond to vertices of a regular convex

n

-gon. We characterize positive integers

n

for which regular drawings of

K_{n}

contain a face forming a convex 5-gon.

To our knowledge, this type of problems has not been considered in the literature before and so we also pose several new natural open problems.

我们开始研究Kn的凸直线图中关于面的极值问题，即顶点由平面上凸位置的点表示，边缘由代表端点的点之间的线段表示的图。我们证明，如果Kn的凸直线图不包含至少三条边的公共内点，则总有一个面形成凸5-gon，而存在这样的图，没有任何面形成k≥6的凸k-gon。如果一个Kn的凸直线图的顶点对应于一个正则凸n-gon的顶点，那么它就是正则的。我们描述正整数n，其中Kn的正则图包含一个形成凸5-gon的面。据我们所知，这类问题在以前的文献中没有被考虑过，所以我们也提出了几个新的自然开放问题。

{"title":"Faces in rectilinear drawings of complete graphs","authors":"Martin Balko , Anna Brötzner , Fabian Klute , Josef Tkadlec","doi":"10.1016/j.ejc.2025.104217","DOIUrl":"10.1016/j.ejc.2025.104217","url":null,"abstract":"<div><div>We initiate the study of extremal problems about faces in convex rectilinear drawings of <math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></math>, that is, drawings where vertices are represented by points in the plane in convex position and edges by line segments between the points representing the end-vertices. We show that if a convex rectilinear drawing of <math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></math> does not contain a common interior point of at least three edges, then there is always a face forming a convex 5-gon while there are such drawings without any face forming a convex <math><mi>k</mi></math>-gon with <math><mrow><mi>k</mi><mo>≥</mo><mn>6</mn></mrow></math>.</div><div>A convex rectilinear drawing of <math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></math> is regular if its vertices correspond to vertices of a regular convex <math><mi>n</mi></math>-gon. We characterize positive integers <math><mi>n</mi></math> for which regular drawings of <math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></math> contain a face forming a convex 5-gon.</div><div>To our knowledge, this type of problems has not been considered in the literature before and so we also pose several new natural open problems.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"130 ","pages":"Article 104217"},"PeriodicalIF":1.0,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144596135","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

A decomposition of cylindric partitions and cylindric partitions into distinct parts 将圆柱分区和圆柱分区分解成不同的部分

IF 1 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2025-12-01 Epub Date: 2025-07-11 DOI: 10.1016/j.ejc.2025.104219

Kağan Kurşungöz, Halı̇me Ömrüuzun Seyrek

We introduce the notion of pivot in a chain of skew diagrams in the context of cylindric partitions. Then, we show that cylindric partitions are in one-to-one correspondence with a pair consisting of an ordinary partition and a suitably restricted chain of pivots. Next, we show the general form of the generating function for cylindric partitions into distinct parts and give some examples. We prove part of a conjecture by Corteel, Dousse, and Uncu. The approaches and proofs are elementary and combinatorial.

我们在圆柱分区的背景下引入了斜图链中的枢轴的概念。然后，我们证明了圆柱分区与一个普通分区和一个适当限制的轴链组成的一对是一一对应的。接下来，我们给出了圆柱划分成不同部分的生成函数的一般形式，并给出了一些例子。我们证明了Corteel， Dousse和Uncu的部分猜想。方法和证明是基本的和组合的。

引用次数: 0

Partitioning 2-edge-coloured bipartite graphs into monochromatic cycles 2边彩色二部图划分为单色圈

IF 1 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2025-12-01 Epub Date: 2025-06-09 DOI: 10.1016/j.ejc.2025.104192

Fabrício Siqueira Benevides , Arthur Lima Quintino , Alexandre Talon

Given an

r

-edge-colouring of the edges of a graph

G

, we say that it can be partitioned into

p

monochromatic cycles when there exists a set of

p

vertex-disjoint monochromatic cycles covering all the vertices of

G

. In the literature of this problem, an edge and a single vertex both count as a cycle.

We show that for every 2-colouring of the edges of a complete balanced bipartite graph,

K_{n, n}

, it can be partitioned into at most 4 monochromatic cycles. This type of question was first studied in 1970 for complete graphs and in 1983, by Gyárfás and Lehel, for

K_{n, n}

. In 2014, Pokrovskiy, showed for all

n

that given any 2-colouring of its edges,

K_{n, n}

can be partitioned into at most three monochromatic paths. It turns out that finding monochromatic cycles instead of paths is a natural question that has also been raised for other graphs. In 2015, Schaudt and Stein showed that 14 cycles are sufficient for sufficiently large 2-edge-coloured

K_{n, n}

给定图G的边的r边着色，当存在p个顶点不相交的单色环的集合覆盖G的所有顶点时，我们说它可以划分为p个单色环。在这个问题的文献中，一条边和一个顶点都算作一个环。我们证明了对于完全平衡二部图Kn，n的每条边的2着色，它可以被划分为最多4个单色环。1970年，Gyárfás和Lehel首次对完全图研究了这类问题，1983年，他们研究了Kn，n。2014年，Pokrovskiy证明了对于所有n，给定其任意两种颜色的边，Kn，n可以划分为最多3条单色路径。事实证明，寻找单色环而不是路径是一个很自然的问题，这个问题在其他图中也被提出过。2015年，Schaudt和Stein证明，对于足够大的2边彩色Kn，n， 14个环是足够的。

{"title":"Partitioning 2-edge-coloured bipartite graphs into monochromatic cycles","authors":"Fabrício Siqueira Benevides , Arthur Lima Quintino , Alexandre Talon","doi":"10.1016/j.ejc.2025.104192","DOIUrl":"10.1016/j.ejc.2025.104192","url":null,"abstract":"<div><div>Given an <math><mi>r</mi></math>-edge-colouring of the edges of a graph <math><mi>G</mi></math>, we say that it can be partitioned into <math><mi>p</mi></math> monochromatic cycles when there exists a set of <math><mi>p</mi></math> vertex-disjoint monochromatic cycles covering all the vertices of <math><mi>G</mi></math>. In the literature of this problem, an edge and a single vertex both count as a cycle.</div><div>We show that for every 2-colouring of the edges of a complete balanced bipartite graph, <math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>n</mi></mrow></msub></math>, it can be partitioned into at most 4 monochromatic cycles. This type of question was first studied in 1970 for complete graphs and in 1983, by Gyárfás and Lehel, for <math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>n</mi></mrow></msub></math>. In 2014, Pokrovskiy, showed for all <math><mi>n</mi></math> that given any 2-colouring of its edges, <math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>n</mi></mrow></msub></math> can be partitioned into at most three monochromatic paths. It turns out that finding monochromatic cycles instead of paths is a natural question that has also been raised for other graphs. In 2015, Schaudt and Stein showed that 14 cycles are sufficient for sufficiently large 2-edge-coloured <math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>n</mi></mrow></msub></math>.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"130 ","pages":"Article 104192"},"PeriodicalIF":1.0,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144241204","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Borsuk and Vázsonyi problems through Reuleaux polyhedra Borsuk和Vázsonyi问题通过勒洛多面体

IF 1 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2025-12-01 Epub Date: 2025-07-12 DOI: 10.1016/j.ejc.2025.104215

Gyivan Lopez-Campos , Déborah Oliveros , Jorge L. Ramírez Alfonsín

The Borsuk conjecture and the Vázsonyi problem are two attractive and famous questions in discrete and combinatorial geometry, both based on the notion of diameter of bounded sets. In this paper, we present an equivalence between the critical sets with Borsuk number 4 in

R^{3}

and the minimal structures for the Vázsonyi problem by using the well-known Reuleaux polyhedra. The latter leads to a full characterization of all finite sets in

R^{3}

with Borsuk number 4.

The proof of such equivalence needs various ingredients, in particular, we proved a conjecture dealing with strongly critical configuration for the Vázsonyi problem and showed that the diameter graph arising from involutive polyhedra is vertex (and edge) 4-critical.

Borsuk猜想和Vázsonyi问题是离散几何和组合几何中两个引人注目的著名问题，它们都基于有界集直径的概念。本文利用著名的勒洛多面体，给出了Vázsonyi问题在R3中Borsuk数为4的临界集与最小结构的等价性。后者导致了R3中具有Borsuk数4的所有有限集的完整表征。这种等价性的证明需要多种成分，特别是我们证明了Vázsonyi问题的一个处理强临界构形的猜想，并证明了对合多面体产生的直径图是顶点（和边）4临界的。

引用次数: 0

Partial-dual genus polynomial of graphs 图的偏对偶格多项式

IF 1 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2025-12-01 Epub Date: 2025-07-09 DOI: 10.1016/j.ejc.2025.104221

Zhiyun Cheng

Recently, Chmutov introduced the partial duality of ribbon graphs, which can be regarded as a generalization of the classical Euler-Poincaré duality. The partial-dual genus polynomial

^{\partial} ɛ_{G} (z)

is an enumeration of the partial duals of

G

by Euler genus. For an intersection graph derived from a given chord diagram, the partial-dual genus polynomial can be defined by considering the ribbon graph associated to the chord diagram. In this paper, we provide a combinatorial approach to the partial-dual genus polynomial in terms of intersection graphs without referring to chord diagrams. After extending the definition of the partial-dual genus polynomial from intersection graphs to all graphs, we prove that it satisfies the four-term relation of graphs. This provides an answer to a problem proposed by Chmutov (2023).

最近，Chmutov引入了带状图的部分对偶性，它可以看作是经典欧拉-庞卡罗对偶性的推广。偏对偶格多项式∂o G(z)是G的偏对偶的欧拉格的枚举。对于由弦图导出的交点图，可以通过考虑与弦图相关联的带状图来定义部分对偶格多项式。在本文中，我们提供了一种不用弦图而用交图表示的部分对偶格多项式的组合方法。将部分对偶格多项式的定义从交图推广到所有图，证明了它满足图的四项关系。这为Chmutov（2023）提出的问题提供了答案。

引用次数: 0

Decomposition of triangle-free planar graphs 无三角形平面图的分解

IF 0.9 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2025-12-01 Epub Date: 2025-08-19 DOI: 10.1016/j.ejc.2025.104227

Rongxing Xu , Xuding Zhu

A decomposition of a graph

G

is a family of subgraphs of

G

whose edge sets form a partition of

E (G)

. In this paper, we prove that every triangle-free planar graph

G

can be decomposed into a 2-degenerate graph and a matching. Consequently, every triangle-free planar graph

G

has a matching

M

such that

G - M

is online 3-DP-colorable. This strengthens an earlier result in Škrekovski (1999) that every triangle-free planar graph is 1-defective 3-choosable.

图G的分解是G的一组子图，这些子图的边集构成E(G)的一个划分。本文证明了每一个无三角形平面图G都可以分解为一个2-简并图和一个匹配图。因此，每一个无三角形平面图G都有一个匹配的M，使得G−M是在线3- dp可着色的。这加强了Škrekovski（1999）中先前的一个结果，即每个无三角形平面图都是1-缺陷3-可选的。

引用次数: 0

Decomposing random regular graphs into stars 将随机规则图分解成星形

IF 1 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics

Pub Date : 2025-12-01 Epub Date: 2025-07-16 DOI: 10.1016/j.ejc.2025.104216

Michelle Delcourt , Catherine Greenhill , Mikhail Isaev , Bernard Lidický , Luke Postle

We study

k

-star decompositions, that is, partitions of the edge set into disjoint stars with

k

edges, in the uniformly random

d

-regular graph model

G_{n, d}

. Using the small subgraph conditioning method, we prove an existence result for such decompositions for all

d, k

such that

d / 2 < k \leq d / 2 + max {1, \frac{1}{6} log d}

. More generally, we give a sufficient existence condition that can be checked numerically for any given values of

d

and

k

. Complementary negative results are obtained using the independence ratio of random regular graphs. Our results establish an existence threshold for

k

-star decompositions in

G_{n, d}

for all

d \leq 100

and

k > d / 2

For smaller values of

k

, the connection between

k

-star decompositions and

β

-orientations allows us to apply results of Thomassen (2012) and Lovász et al. (2013). We prove that random

d

-regular graphs satisfy their assumptions with high probability, thus establishing a.a.s. existence of

k

-star decompositions (i) when

2 k^{2} + k \leq d

, and (ii) when

k

is odd and

k < d / 2

我们研究了均匀随机d规则图模型Gn，d中的k星分解，即将边集划分为具有k条边的不相交星。利用小子图条件法，证明了d，k的所有分解的存在性，使得d/2<；k≤d/2+max{1,16logd}。更一般地，我们给出了对于任意给定的d和k值都可以用数值检验的充分存在性条件。利用随机正则图的独立比得到了互补的负结果。我们的结果为Gn、d中所有d≤100和k>；d/2的k星分解建立了存在阈值。对于较小的k值，k星分解与β取向之间的联系使我们能够应用Thomassen（2012）和Lovász等人（2013）的结果。我们证明了随机d正则图高概率地满足它们的假设，从而建立了k-星分解(i)当2k2+k≤d，以及（ii）当k为奇数且k<；d/2时的a.a.s.存在性。

{"title":"Decomposing random regular graphs into stars","authors":"Michelle Delcourt , Catherine Greenhill , Mikhail Isaev , Bernard Lidický , Luke Postle","doi":"10.1016/j.ejc.2025.104216","DOIUrl":"10.1016/j.ejc.2025.104216","url":null,"abstract":"<div><div>We study <math><mi>k</mi></math>-star decompositions, that is, partitions of the edge set into disjoint stars with <math><mi>k</mi></math> edges, in the uniformly random <math><mi>d</mi></math>-regular graph model <math><msub><mrow><mi>G</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>d</mi></mrow></msub></math>. Using the small subgraph conditioning method, we prove an existence result for such decompositions for all <math><mrow><mi>d</mi><mo>,</mo><mi>k</mi></mrow></math> such that <math><mrow><mi>d</mi><mo>/</mo><mn>2</mn><mo><</mo><mi>k</mi><mo>≤</mo><mi>d</mi><mo>/</mo><mn>2</mn><mo>+</mo><mo>max</mo><mrow><mo>{</mo><mn>1</mn><mo>,</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>6</mn></mrow></mfrac><mo>log</mo><mi>d</mi><mo>}</mo></mrow></mrow></math>. More generally, we give a sufficient existence condition that can be checked numerically for any given values of <math><mi>d</mi></math> and <math><mi>k</mi></math>. Complementary negative results are obtained using the independence ratio of random regular graphs. Our results establish an existence threshold for <math><mi>k</mi></math>-star decompositions in <math><msub><mrow><mi>G</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>d</mi></mrow></msub></math> for all <math><mrow><mi>d</mi><mo>≤</mo><mn>100</mn></mrow></math> and <math><mrow><mi>k</mi><mo>></mo><mi>d</mi><mo>/</mo><mn>2</mn></mrow></math>.</div><div>For smaller values of <math><mi>k</mi></math>, the connection between <math><mi>k</mi></math>-star decompositions and <math><mi>β</mi></math>-orientations allows us to apply results of Thomassen (2012) and Lovász et al. (2013). We prove that random <math><mi>d</mi></math>-regular graphs satisfy their assumptions with high probability, thus establishing a.a.s. existence of <math><mi>k</mi></math>-star decompositions (i) when <math><mrow><mn>2</mn><msup><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mi>k</mi><mo>≤</mo><mi>d</mi></mrow></math>, and (ii) when <math><mi>k</mi></math> is odd and <math><mrow><mi>k</mi><mo><</mo><mi>d</mi><mo>/</mo><mn>2</mn></mrow></math>.</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"130 ","pages":"Article 104216"},"PeriodicalIF":1.0,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144634400","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

首页上一页

类型

全部化学•材料生命科学医学物理工程技术环境•农林材料科学地球科学法学管理学化学环境科学与生态学计算机科学教育学经济学农林科学人文科学生物学数学物理与天体物理心理学综合性期刊其他工业工程理学历史学农学文学信息工程

数据库

全部 ACS Publications Elsevier ieeexplore Springer The Royal Society of Chemistry Wiley

期刊

European Journal of Combinatorics

全部 Acc. Chem. Res. ACS Applied Bio Materials ACS Appl. Electron. Mater. ACS Appl. Energy Mater. ACS Appl. Mater. Interfaces ACS Appl. Nano Mater. ACS Appl. Polym. Mater. ACS BIOMATER-SCI ENG ACS Catal. ACS Cent. Sci. ACS Chem. Biol. ACS Chemical Health & Safety ACS Chem. Neurosci. ACS Comb. Sci. ACS Earth Space Chem. ACS Energy Lett. ACS Infect. Dis. ACS Macro Lett. ACS Mater. Lett. ACS Med. Chem. Lett. ACS Nano ACS Omega ACS Photonics ACS Sens. ACS Sustainable Chem. Eng. ACS Synth. Biol. Anal. Chem. BIOCHEMISTRY-US Bioconjugate Chem. BIOMACROMOLECULES Chem. Res. Toxicol. Chem. Rev. Chem. Mater. CRYST GROWTH DES ENERG FUEL Environ. Sci. Technol. Environ. Sci. Technol. Lett. Eur. J. Inorg. Chem. IND ENG CHEM RES Inorg. Chem. J. Agric. Food. Chem. J. Chem. Eng. Data J. Chem. Educ. J. Chem. Inf. Model. J. Chem. Theory Comput. J. Med. Chem. J. Nat. Prod. J PROTEOME RES J. Am. Chem. Soc. LANGMUIR MACROMOLECULES Mol. Pharmaceutics Nano Lett. Org. Lett. ORG PROCESS RES DEV ORGANOMETALLICS J. Org. Chem. J. Phys. Chem. J. Phys. Chem. A J. Phys. Chem. B J. Phys. Chem. C J. Phys. Chem. Lett. Analyst Anal. Methods Biomater. Sci. Catal. Sci. Technol. Chem. Commun. Chem. Soc. Rev. CHEM EDUC RES PRACT CRYSTENGCOMM Dalton Trans. Energy Environ. Sci. ENVIRON SCI-NANO ENVIRON SCI-PROC IMP ENVIRON SCI-WAT RES Faraday Discuss. Food Funct. Green Chem. Inorg. Chem. Front. Integr. Biol. J. Anal. At. Spectrom. J. Mater. Chem. A J. Mater. Chem. B J. Mater. Chem. C Lab Chip Mater. Chem. Front. Mater. Horiz. MEDCHEMCOMM Metallomics Mol. Biosyst. Mol. Syst. Des. Eng. Nanoscale Nanoscale Horiz. Nat. Prod. Rep. New J. Chem. Org. Biomol. Chem. Org. Chem. Front. PHOTOCH PHOTOBIO SCI PCCP Polym. Chem.

﹀