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Journal of Combinatorial Theory Series B最新文献

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Discrepancy and sparsity 差异和稀疏性
IF 1.2 1区 数学 Q1 MATHEMATICS Pub Date : 2024-06-20 DOI: 10.1016/j.jctb.2024.06.001
Mario Grobler , Yiting Jiang , Patrice Ossona de Mendez , Sebastian Siebertz , Alexandre Vigny

We study the connections between the notions of combinatorial discrepancy and graph degeneracy. In particular, we prove that the maximum discrepancy over all subgraphs H of a graph G of the neighborhood set system of H is sandwiched between Ω(logdeg(G)) and O(deg(G)), where deg(G) denotes the degeneracy of G. We extend this result to inequalities relating weak coloring numbers and discrepancy of graph powers and deduce a new characterization of bounded expansion classes.

Then we switch to a model theoretical point of view, introduce pointer structures, and study their relations to graph classes with bounded expansion. We deduce that a monotone class of graphs has bounded expansion if and only if all the set systems definable in this class have bounded hereditary discrepancy.

Using known bounds on the VC-density of set systems definable in nowhere dense classes we also give a characterization of nowhere dense classes in terms of discrepancy.

As consequences of our results, we obtain a corollary on the discrepancy of neighborhood set systems of edge colored graphs, a polynomial-time algorithm to compute ε-approximations of size O(1/ε) for set systems definable in bounded expansion classes, an application to clique coloring, and even the non-existence of a quantifier elimination scheme for nowhere dense classes.

我们研究了组合差异和图退化概念之间的联系。特别是,我们证明了 H 的邻集系统图 G 的所有子图 H 的最大差异介于 Ω(logdeg(G)) 和 O(deg(G)) 之间,其中 deg(G) 表示 G 的退化度。我们将这一结果扩展到与弱着色数和图幂差异有关的不等式,并推导出有界扩展类的新特征。然后,我们转换到模型理论的视角,引入指针结构,并研究它们与有界扩展图类的关系。我们推导出,当且仅当一个单调图类中所有可定义的集合系统都具有有界遗传差异时,该类才具有有界扩展。作为我们结果的后果,我们得到了关于边缘着色图的邻域集合系统差异的推论、计算可定义在有界扩展类中的集合系统的大小为 O(1/ε)的ε近似的多项式时间算法、对簇着色的应用,甚至无处密集类的量词消除方案的不存在。
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引用次数: 0
On the use of senders for asymmetric tuples of cliques in Ramsey theory 论拉姆齐理论中不对称元组小群的发送者使用问题
IF 1.4 1区 数学 Q1 MATHEMATICS Pub Date : 2024-06-18 DOI: 10.1016/j.jctb.2024.05.006
Simona Boyadzhiyska , Thomas Lesgourgues

A graph G is q-Ramsey for a q-tuple of graphs (H1,,Hq) if for every q-coloring of the edges of G there exists a monochromatic copy of Hi in color i for some i[q]. Over the last few decades, researchers have investigated a number of questions related to this notion, aiming to understand the properties of graphs that are q-Ramsey for a fixed tuple. Among the tools developed while studying questions of this type are gadget graphs, called signal senders and determiners, which have proven invaluable for building Ramsey graphs with certain properties. However, until now these gadgets have been shown to exist and used mainly in the two-color setting or in the symmetric multicolor setting, and our knowledge about their existence for multicolor asymmetric tuples is extremely limited. In this paper, we construct such gadgets for any tuple of cliques. We then use these gadgets to generalize three classical theorems in this area to the asymmetric multicolor setting.

如果对于 G 的边的每 q 种颜色,在某个 i∈[q]中都存在 Hi 的单色副本,那么对于图的 q 组(H1,...,Hq)来说,图 G 是 q-Ramsey 图。在过去的几十年里,研究人员研究了许多与这一概念相关的问题,旨在了解对于固定元组而言具有 q-Ramsey 的图的性质。在研究这类问题的过程中开发的工具包括小工具图,即信号发送器和确定器,它们已被证明在构建具有某些属性的拉姆齐图时非常有用。然而,到目前为止,这些小工具主要是在双色或对称多色环境中被证明存在和使用,而我们对多色非对称图元存在的了解极为有限。在本文中,我们为任何元组构建了这种小工具。然后,我们利用这些小工具将这一领域的三个经典定理推广到非对称多色环境中。
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引用次数: 0
On the difference of mean subtree orders under edge contraction 论边缘收缩下平均子树顺序的差异
IF 1.4 1区 数学 Q1 MATHEMATICS Pub Date : 2024-06-18 DOI: 10.1016/j.jctb.2024.06.002
Ruoyu Wang

Given a tree T of order n, one can contract any edge and obtain a new tree T of order n1. In 1983, Jamison made a conjecture that the mean subtree order, i.e., the average order of all subtrees, decreases at least 13 in contracting an edge of a tree. In 2023, Luo, Xu, Wagner and Wang proved the case when the edge to be contracted is a pendant edge. In this article, we prove that the conjecture is true in general.

给定一棵阶数为 n 的树 T,可以收缩任意一条边,得到一棵阶数为 n-1 的新树 T⁎。1983 年,Jamison 提出了一个猜想,即在收缩树的一条边时,平均子树序(即所有子树的平均序)至少会减少 13。2023 年,Luo、Xu、Wagner 和 Wang 证明了要收缩的边是垂边时的情况。在本文中,我们将证明该猜想在一般情况下为真。
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引用次数: 0
Locally finite vertex-rotary maps and coset graphs with finite valency and finite edge multiplicity 局部有限顶点旋转映射和具有有限价及有限边乘数的余集图
IF 1.4 1区 数学 Q1 MATHEMATICS Pub Date : 2024-06-11 DOI: 10.1016/j.jctb.2024.05.005
Cai Heng Li , Cheryl E. Praeger , Shu Jiao Song

A well-known theorem of Sabidussi shows that a simple G-arc-transitive graph can be represented as a coset graph for the group G. This pivotal result is the standard way to turn problems about simple arc-transitive graphs into questions about groups. In this paper, the Sabidussi representation is extended to arc-transitive, not necessarily simple graphs which satisfy a local-finiteness condition: namely graphs with finite valency and finite edge-multiplicity. The construction yields a G-arc-transitive coset graph Cos(G,H,J), where H,J are stabilisers in G of a vertex and incident edge, respectively. A first major application is presented concerning arc-transitive maps on surfaces: given a group G=a,z with |z|=2 and |a| finite, the coset graph Cos(G,a,z) is shown, under suitable finiteness assumptions, to have exactly two different arc-transitive embeddings as a G-arc-transitive map (V,E,F) (with V,E,F the sets of vertices, edges and faces, respectively), namely, a G-rotary map if |az| is finite, and a G-bi-rotary map if |zza| is finite. The G-rotary map can be represented as a coset geometry for G, extending the notion of a coset graph. However the G-bi-rotary map does not have such a representation, and the face boundary cycles must be specified in addition to incidences between faces and edges. In addition a coset geometry construction is given of a flag-regular map (V,E,F) for non necessarily simple graphs. For all of these constructions it is proved that the face boundary cycles are simple cycles precisely when the given group acts faithfully on VF. Illustrative examples are given for graphs related to the n-dimensional hypercubes and the Petersen graph.

萨比杜西(Sabidussi)的一个著名定理表明,简单的 G-弧透图可以表示为群 G 的余集图。这一关键结果是将简单弧透图问题转化为群问题的标准方法。在本文中,萨比杜西表示法被扩展到了满足局部有限性条件的弧遍历图,而不一定是简单图:即具有有限价和有限边多重性的图。该构造产生了一个 G-弧遍历余集图 Cos(G,H,J),其中 H,J 分别是顶点和入射边在 G 中的稳定器。本文提出的第一个主要应用涉及曲面上的弧跨映射:给定一个组 G=〈a,z〉,|z|=2,|a|有限,在适当的有限性假设下,证明了余集图 Cos(G,〈a〉,〈z〉) 作为 G-弧透映射 (V. E,F) 有两种不同的弧透嵌入、E,F)(V,E,F 分别为顶点集、边集和面集),即如果 |az| 有限,则为 G 旋转图;如果 |zza| 有限,则为 G 双旋转图。G 旋转图可以表示为 G 的余集几何,扩展了余集图的概念。然而 G-bi-rotary 映射没有这样的表示法,除了面与边之间的发生率之外,还必须指定面边界循环。此外,还给出了非简单图的旗正则图(V,E,F)的余集几何构造。对于所有这些构造,都证明了当给定的群忠实地作用于 V∪F 时,面边界循环正是简单循环。文中给出了与 n 维超立方体和彼得森图有关的图的示例。
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引用次数: 0
Counting oriented trees in digraphs with large minimum semidegree 在具有大最小半度的图中计算定向树
IF 1.4 1区 数学 Q1 MATHEMATICS Pub Date : 2024-05-29 DOI: 10.1016/j.jctb.2024.05.004
Felix Joos, Jonathan Schrodt

Let T be an oriented tree on n vertices with maximum degree at most eo(logn). If G is a digraph on n vertices with minimum semidegree δ0(G)(12+o(1))n, then G contains T as a spanning tree, as recently shown by Kathapurkar and Montgomery (in fact, they only require maximum degree o(n/logn)). This generalizes the corresponding result by Komlós, Sárközy and Szemerédi for graphs. We investigate the natural question how many copies of T the digraph G contains. Our main result states that every such G contains at least |Aut(T)|1(12o(1))nn! copies of T, which is optimal. This implies the analogous result in the undirected case.

设 T 是 n 个顶点上的定向树,其最大度最多为 eo(logn)。如果 G 是 n 个顶点上的数图,最小半度 δ0(G)≥(12+o(1))n,那么 G 包含作为生成树的 T,正如 Kathapurkar 和 Montgomery 最近证明的那样(事实上,他们只要求最大度为 o(n/logn))。这推广了 Komlós、Sárközy 和 Szemerédi 对图的相应结果。我们研究了数图 G 包含多少份 T 的自然问题。我们的主要结果表明,每个这样的 G 至少包含 T 的 |Aut(T)|-1(12-o(1))nn!这意味着无向情况下的类似结果。
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引用次数: 0
The burning number conjecture holds asymptotically 燃烧数猜想近似成立
IF 1.4 1区 数学 Q1 MATHEMATICS Pub Date : 2024-05-29 DOI: 10.1016/j.jctb.2024.05.003
Sergey Norin, Jérémie Turcotte

The burning number b(G) of a graph G is the smallest number of turns required to burn all vertices of a graph if at every turn a new fire is started and existing fires spread to all adjacent vertices. The Burning Number Conjecture of Bonato et al. (2016) postulates that b(G)n for all connected graphs G on n vertices. We prove that this conjecture holds asymptotically, that is b(G)(1+o(1))n.

图 G 的燃烧数 b(G)是指如果每转一圈都有新的火开始燃烧,并且已有的火蔓延到所有相邻的顶点,则烧毁图中所有顶点所需的最小圈数。Bonato 等人(2016 年)提出的燃烧次数猜想假设,对于 n 个顶点上的所有连通图 G,b(G)≤⌈n⌉。我们证明这一猜想近似成立,即 b(G)≤(1+o(1))n。
{"title":"The burning number conjecture holds asymptotically","authors":"Sergey Norin,&nbsp;Jérémie Turcotte","doi":"10.1016/j.jctb.2024.05.003","DOIUrl":"https://doi.org/10.1016/j.jctb.2024.05.003","url":null,"abstract":"<div><p>The burning number <span><math><mi>b</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> of a graph <em>G</em> is the smallest number of turns required to burn all vertices of a graph if at every turn a new fire is started and existing fires spread to all adjacent vertices. The Burning Number Conjecture of Bonato et al. (2016) postulates that <span><math><mi>b</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mrow><mo>⌈</mo><msqrt><mrow><mi>n</mi></mrow></msqrt><mo>⌉</mo></mrow></math></span> for all connected graphs <em>G</em> on <em>n</em> vertices. We prove that this conjecture holds asymptotically, that is <span><math><mi>b</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mo>(</mo><mn>1</mn><mo>+</mo><mi>o</mi><mo>(</mo><mn>1</mn><mo>)</mo><mo>)</mo><msqrt><mrow><mi>n</mi></mrow></msqrt></math></span>.</p></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"168 ","pages":"Pages 208-235"},"PeriodicalIF":1.4,"publicationDate":"2024-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141244764","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"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
Directed cycles with zero weight in Zpk Zpk 中权重为零的有向循环
IF 1.4 1区 数学 Q1 MATHEMATICS Pub Date : 2024-05-29 DOI: 10.1016/j.jctb.2024.05.002
Shoham Letzter , Natasha Morrison

For a finite abelian group A, define f(A) to be the minimum integer such that for every complete digraph Γ on f vertices and every map w:E(Γ)A, there exists a directed cycle C in Γ such that eE(C)w(e)=0. The study of f(A) was initiated by Alon and Krivelevich (2021). In this article, we prove that f(Zpk)=O(pk(logk)2), where p is prime, with an improved bound of O(klogk) when p=2. These bounds are tight up to a factor which is polylogarithmic in k.

对于有限无边群 A,定义 f(A) 为最小整数,即对于 f 个顶点上的每个完整图 Γ 和每个映射 w:E(Γ)→A, Γ 中存在一个有向循环 C,使得∑e∈E(C)w(e)=0。 f(A) 的研究由 Alon 和 Krivelevich (2021) 发起。在这篇文章中,我们证明了 f(Zpk)=O(pk(logk)2),其中 p 是素数,当 p=2 时的改进边界为 O(klogk)。这些界值在 k 的多对数因子以内都很紧。
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引用次数: 0
Dirac-type theorems for long Berge cycles in hypergraphs 超图中长 Berge 循环的狄拉克型定理
IF 1.4 1区 数学 Q1 MATHEMATICS Pub Date : 2024-05-22 DOI: 10.1016/j.jctb.2024.05.001
Alexandr Kostochka , Ruth Luo , Grace McCourt

The famous Dirac's Theorem gives an exact bound on the minimum degree of an n-vertex graph guaranteeing the existence of a hamiltonian cycle. In the same paper, Dirac also observed that a graph with minimum degree at least k2 contains a cycle of length at least k+1. The purpose of this paper is twofold: we prove exact bounds of similar type for hamiltonian Berge cycles as well as for Berge cycles of length at least k in r-uniform, n-vertex hypergraphs for all combinations of k,r and n with 3r,kn. The bounds differ for different ranges of r compared to n and k.

著名的狄拉克定理给出了 n 个顶点图的最小度的精确约束,保证了哈密顿循环的存在。在同一篇文章中,狄拉克还观察到一个最小度至少为 k≥2 的图包含一个长度至少为 k+1 的循环。本文的目的有两个:我们证明了类似类型的哈密顿贝格循环以及长度至少为 k 的 r-uniform n 顶点超图中的贝格循环的精确边界,适用于 3≤r,k≤n 的 k、r 和 n 的所有组合。与 n 和 k 相比,r 的范围不同,界限也不同。
{"title":"Dirac-type theorems for long Berge cycles in hypergraphs","authors":"Alexandr Kostochka ,&nbsp;Ruth Luo ,&nbsp;Grace McCourt","doi":"10.1016/j.jctb.2024.05.001","DOIUrl":"https://doi.org/10.1016/j.jctb.2024.05.001","url":null,"abstract":"<div><p>The famous Dirac's Theorem gives an exact bound on the minimum degree of an <em>n</em>-vertex graph guaranteeing the existence of a hamiltonian cycle. In the same paper, Dirac also observed that a graph with minimum degree at least <span><math><mi>k</mi><mo>≥</mo><mn>2</mn></math></span> contains a cycle of length at least <span><math><mi>k</mi><mo>+</mo><mn>1</mn></math></span>. The purpose of this paper is twofold: we prove exact bounds of similar type for hamiltonian Berge cycles as well as for Berge cycles of length at least <em>k</em> in <em>r</em>-uniform, <em>n</em>-vertex hypergraphs for all combinations of <span><math><mi>k</mi><mo>,</mo><mi>r</mi></math></span> and <em>n</em> with <span><math><mn>3</mn><mo>≤</mo><mi>r</mi><mo>,</mo><mi>k</mi><mo>≤</mo><mi>n</mi></math></span>. The bounds differ for different ranges of <em>r</em> compared to <em>n</em> and <em>k</em>.</p></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"168 ","pages":"Pages 159-191"},"PeriodicalIF":1.4,"publicationDate":"2024-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141078547","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"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
Graphs with all holes the same length 所有孔长度相同的图形
IF 1.4 1区 数学 Q1 MATHEMATICS Pub Date : 2024-05-14 DOI: 10.1016/j.jctb.2024.04.006
Linda Cook , Jake Horsfield , Myriam Preissmann , Cléophée Robin , Paul Seymour , Ni Luh Dewi Sintiari , Nicolas Trotignon , Kristina Vušković

A graph is ℓ-holed if all its induced cycles of length at least four have length exactly . We give a complete description of the -holed graphs for each 7.

如果一个图的所有长度至少为四的诱导循环的长度正好为 ℓ,那么这个图就是 ℓ-holed 图。我们给出了每个 ℓ≥7 的 ℓ-holed 图的完整描述。
{"title":"Graphs with all holes the same length","authors":"Linda Cook ,&nbsp;Jake Horsfield ,&nbsp;Myriam Preissmann ,&nbsp;Cléophée Robin ,&nbsp;Paul Seymour ,&nbsp;Ni Luh Dewi Sintiari ,&nbsp;Nicolas Trotignon ,&nbsp;Kristina Vušković","doi":"10.1016/j.jctb.2024.04.006","DOIUrl":"https://doi.org/10.1016/j.jctb.2024.04.006","url":null,"abstract":"<div><p>A graph is <em>ℓ-holed</em> if all its induced cycles of length at least four have length exactly <em>ℓ</em>. We give a complete description of the <em>ℓ</em>-holed graphs for each <span><math><mi>ℓ</mi><mo>≥</mo><mn>7</mn></math></span>.</p></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"168 ","pages":"Pages 96-158"},"PeriodicalIF":1.4,"publicationDate":"2024-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140924613","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"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
No perfect state transfer in trees with more than 3 vertices 有 3 个以上顶点的树中没有完美的状态转移
IF 1.4 1区 数学 Q1 MATHEMATICS Pub Date : 2024-05-10 DOI: 10.1016/j.jctb.2024.04.004
Gabriel Coutinho, Emanuel Juliano, Thomás Jung Spier

We prove that the only trees that admit perfect state transfer according to the adjacency matrix model are P2 and P3. This answers a question first asked by Godsil in 2012 and proves a conjecture by Coutinho and Liu from 2015.

我们证明,根据邻接矩阵模型,只有 P2 和 P3 树可以实现完美的状态转移。这回答了 Godsil 在 2012 年首次提出的问题,并证明了 Coutinho 和 Liu 在 2015 年提出的猜想。
{"title":"No perfect state transfer in trees with more than 3 vertices","authors":"Gabriel Coutinho,&nbsp;Emanuel Juliano,&nbsp;Thomás Jung Spier","doi":"10.1016/j.jctb.2024.04.004","DOIUrl":"https://doi.org/10.1016/j.jctb.2024.04.004","url":null,"abstract":"<div><p>We prove that the only trees that admit perfect state transfer according to the adjacency matrix model are <span><math><msub><mrow><mi>P</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>P</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span>. This answers a question first asked by Godsil in 2012 and proves a conjecture by Coutinho and Liu from 2015.</p></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"168 ","pages":"Pages 68-85"},"PeriodicalIF":1.4,"publicationDate":"2024-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140901144","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"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
期刊
Journal of Combinatorial Theory Series B
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