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Bounded-Diameter Tree-Decompositions 有界直径树分解
IF 1.1 2区 数学 Q1 MATHEMATICS Pub Date : 2024-04-08 DOI: 10.1007/s00493-024-00088-1
Eli Berger, Paul Seymour

When does a graph admit a tree-decomposition in which every bag has small diameter? For finite graphs, this is a property of interest in algorithmic graph theory, where it is called having bounded “tree-length”. We will show that this is equivalent to being “boundedly quasi-isometric to a tree”, which for infinite graphs is a much-studied property from metric geometry. One object of this paper is to tie these two areas together. We will prove that there is a tree-decomposition in which each bag has small diameter, if and only if there is a map (phi ) from V(G) into the vertex set of a tree T, such that for all (u,vin V(G)), the distances (d_G(u,v), d_T(phi (u),phi (v))) differ by at most a constant. A necessary condition for admitting such a tree-decomposition is that there is no long geodesic cycle, and for graphs of bounded tree-width, Diestel and Müller showed that this is also sufficient. But it is not sufficient in general, even qualitatively, because there are graphs in which every geodesic cycle has length at most three, and yet every tree-decomposition has a bag with large diameter. There is a more general necessary condition, however. A “geodesic loaded cycle” in G is a pair (CF), where C is a cycle of G and (Fsubseteq E(C)), such that for every pair uv of vertices of C, one of the paths of C between uv contains at most (d_G(u,v)) F-edges, where (d_G(u,v)) is the distance between uv in G. We will show that a (possibly infinite) graph G admits a tree-decomposition in which every bag has small diameter, if and only if |F| is small for every geodesic loaded cycle (CF). Our proof is an extension of an algorithm to approximate tree-length in finite graphs by Dourisboure and Gavoille. In metric geometry, there is a similar theorem that characterizes when a graph is quasi-isometric to a tree, “Manning’s bottleneck criterion”. The goal of this paper is to tie all these concepts together, and add a few more related ideas. For instance, we prove a conjecture of Rose McCarty, that G admits a tree-decomposition in which every bag has small diameter, if and only if for all vertices uvw of G, some ball of small radius meets every path joining two of uvw.

什么情况下一个图可以进行树形分解,其中每个包的直径都很小?对于有限图,这是算法图论中的一个重要特性,在算法图论中称为有界 "树长"。我们将证明,这等同于 "有界准等距于树",而对于无限图,这是公元几何中研究得很多的一个特性。本文的一个目的是将这两个领域结合起来。我们将证明,当且仅当存在一个从 V(G) 到树 T 的顶点集的映射 (phi ),使得对于 V(G) 中的所有 (u,v),距离 (d_G(u,v), d_T(phi(u),phi(v)))最多相差一个常数时,存在一个树分解,其中每个包都有小直径。对于有界树宽的图,Diestel 和 Müller 证明这也是充分条件。但这在一般情况下是不充分的,甚至在定性上也是不充分的,因为在有些图中,每个大地周期的长度最多只有三个,但每个树分解都有一个大直径的包。不过,还有一个更普遍的必要条件。G 中的 "测地线加载循环 "是指一对 (C, F),其中 C 是 G 的一个循环,并且 (Fsubseteq E(C)), 这样对于 C 的每一对顶点 u, v,C 中 u, v 之间的一条路径最多包含 (d_G(u,v))我们将证明,当且仅当 |F| 对于每个测地线加载的循环 (C, F) 都很小时,一个(可能是无限的)图 G 允许树形分解,其中每个包都有很小的直径。我们的证明是对 Dourisboure 和 Gavoille 提出的有限图中近似树长算法的扩展。在度量几何中,也有一个类似的定理,即 "曼宁瓶颈准则",用来描述一个图何时与一棵树准等距。本文的目的是将所有这些概念联系起来,并增加一些相关的想法。例如,我们证明了罗斯-麦卡蒂(Rose McCarty)的一个猜想,即对于 G 的所有顶点 u、v、w,当且仅当某个小半径球与连接 u、v、w 中两个顶点的每条路径相交时,G 可进行树形分解,其中每个袋的直径都很小。
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引用次数: 0
Induced Subgraphs and Tree Decompositions VIII: Excluding a Forest in (Theta, Prism)-Free Graphs 诱导子图和树分解 VIII:在(Theta、Prism)无图中排除森林
IF 1.1 2区 数学 Q1 MATHEMATICS Pub Date : 2024-04-08 DOI: 10.1007/s00493-024-00097-0
Tara Abrishami, Bogdan Alecu, Maria Chudnovsky, Sepehr Hajebi, Sophie Spirkl

Given a graph H, we prove that every (theta, prism)-free graph of sufficiently large treewidth contains either a large clique or an induced subgraph isomorphic to H, if and only if H is a forest.

给定一个图 H,我们证明,当且仅当 H 是一个森林时,每个具有足够大树宽的(θ,棱)无图都包含一个大簇或一个与 H 同构的诱导子图。
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引用次数: 0
Upper Tail Behavior of the Number of Triangles in Random Graphs with Constant Average Degree 平均度数恒定的随机图中三角形数量的上端行为
IF 1.1 2区 数学 Q1 MATHEMATICS Pub Date : 2024-04-04 DOI: 10.1007/s00493-024-00086-3
Shirshendu Ganguly, Ella Hiesmayr, Kyeongsik Nam

Let N be the number of triangles in an Erdős–Rényi graph ({mathcal {G}}(n,p)) on n vertices with edge density (p=d/n,) where (d>0) is a fixed constant. It is well known that N weakly converges to the Poisson distribution with mean ({d^3}/{6}) as (nrightarrow infty ). We address the upper tail problem for N, namely, we investigate how fast k must grow, so that ({mathbb {P}}(Nge k)) is not well approximated anymore by the tail of the corresponding Poisson variable. Proving that the tail exhibits a sharp phase transition, we essentially show that the upper tail is governed by Poisson behavior only when (k^{1/3} log k< (frac{3}{sqrt{2}} - {o(1)})^{2/3} log n) (sub-critical regime) as well as pin down the tail behavior when (k^{1/3} log k> (frac{3}{sqrt{2}} + {o(1)})^{2/3} log n) (super-critical regime). We further prove a structure theorem, showing that the sub-critical upper tail behavior is dictated by the appearance of almost k vertex-disjoint triangles whereas in the supercritical regime, the excess triangles arise from a clique like structure of size approximately ((6k)^{1/3}). This settles the long-standing upper-tail problem in this case, answering a question of Aldous, complementing a long sequence of works, spanning multiple decades and culminating in Harel et al. (Duke Math J 171(10):2089–2192, 2022), which analyzed the problem only in the regime (pgg frac{1}{n}.) The proofs rely on several novel graph theoretical results which could have other applications.

设 N 是一个厄尔多斯-雷尼图(Erdős-Rényi graph)中三角形的数量,该图有 n 个顶点,边密度为(p=d/n,),其中(d>0)是一个固定常数。众所周知,N弱收敛于均值为({d^3}/{6})的泊松分布。我们要解决 N 的上尾问题,即研究 k 必须增长多快才能使 ({mathbb {P}}(Nge k))不再被相应泊松变量的尾部很好地近似。为了证明尾部会出现急剧的相变,我们从本质上证明,只有当 (k^{1/3}log k< (frac{3}{sqrt{2}}- {o(1)})^{2/3}log k> (frac{3}{sqrt{2}}+ {o(1)})^{2/3}log n) (超临界机制)。我们进一步证明了一个结构定理,表明亚临界上尾行为是由近 k 个顶点相交三角形的出现决定的,而在超临界机制中,多余的三角形来自于一个类似于小集团的结构,其大小约为((6k)^{1/3})。在这种情况下,解决了长期存在的上尾问题,回答了奥尔德斯的一个问题,补充了横跨数十年的一系列工作,并在哈雷尔等人(Duke Math J 171(10):2089-2192,2022)的研究中达到了顶峰,该研究仅在 (pgg frac{1}{n}.) 机制下分析了该问题。
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引用次数: 0
The Number of Topological Types of Trees 树木拓扑类型的数量
IF 1.1 2区 数学 Q1 MATHEMATICS Pub Date : 2024-04-04 DOI: 10.1007/s00493-024-00087-2

Abstract

Two graphs are of the same topological type if they can be mutually embedded into each other topologically. We show that there are exactly (aleph _1) distinct topological types of countable trees. In general, for any infinite cardinal (kappa ) there are exactly (kappa ^+) distinct topological types of trees of size (kappa ) . This solves a problem of van der Holst from 2005.

摘要 如果两个图在拓扑上可以相互嵌入,那么它们就具有相同的拓扑类型。我们证明,可数树的拓扑类型恰好有(aleph _1)种不同的拓扑类型。一般来说,对于任意一个无限红心(kappa ),大小为 (kappa )的树恰好有 (kappa ^+) 个不同的拓扑类型。这解决了 van der Holst 在 2005 年提出的一个问题。
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引用次数: 0
Isoperimetric Inequalities and Supercritical Percolation on High-Dimensional Graphs 高维图上的等周不等式和超临界渗流
IF 1.1 2区 数学 Q1 MATHEMATICS Pub Date : 2024-04-04 DOI: 10.1007/s00493-024-00089-0

Abstract

It is known that many different types of finite random subgraph models undergo quantitatively similar phase transitions around their percolation thresholds, and the proofs of these results rely on isoperimetric properties of the underlying host graph. Recently, the authors showed that such a phase transition occurs in a large class of regular high-dimensional product graphs, generalising a classic result for the hypercube. In this paper we give new isoperimetric inequalities for such regular high-dimensional product graphs, which generalise the well-known isoperimetric inequality of Harper for the hypercube, and are asymptotically sharp for a wide range of set sizes. We then use these isoperimetric properties to investigate the structure of the giant component (L_1) in supercritical percolation on these product graphs, that is, when (p=frac{1+epsilon }{d}) , where d is the degree of the product graph and (epsilon >0) is a small enough constant. We show that typically (L_1) has edge-expansion (Omega left( frac{1}{dln d}right) ) . Furthermore, we show that (L_1) likely contains a linear-sized subgraph with vertex-expansion (Omega left( frac{1}{dln d}right) ) . These results are best possible up to the logarithmic factor in d. Using these likely expansion properties, we determine, up to small polylogarithmic factors in d, the likely diameter of (L_1) as well as the typical mixing time of a lazy random walk on (L_1) . Furthermore, we show the likely existence of a cycle of length (Omega left( frac{n}{dln d}right) ) . These results not only generalise, but also improve substantially upon the known bounds in the case of the hypercube, where in particular the likely diameter and typical mixing time of (L_1) were previously only known to be polynomial in d.

摘要 众所周知,许多不同类型的有限随机子图模型在其渗流阈值附近会发生数量上相似的相变,而这些结果的证明依赖于底层主图的等周特性。最近,作者证明了这种相变发生在一大类规则的高维积图中,推广了超立方体的经典结果。在本文中,我们为这类正则高维积图给出了新的等周不等式,这些等周不等式概括了著名的 Harper 关于超立方体的等周不等式,并且在很宽的集合大小范围内都是渐近尖锐的。然后,我们利用这些等周性质来研究这些积图上超临界渗流中巨型分量 (L_1)的结构,即当 (p=frac{1+epsilon }{d}) 时,其中 d 是积图的度,而 (epsilon>0)是一个足够小的常数。我们证明了典型的 (L_1) 有边扩展 (Omega left( frac{1}{dln d}right) ) 。此外,我们证明了 (L_1) 很可能包含一个线性大小的子图,其顶点展开为 (Omega left(frac{1}{dln d}right) )。利用这些可能的扩展特性,我们确定了(直到 d 的小对数因子)(L_1) 的可能直径以及懒惰随机行走在 (L_1) 上的典型混合时间。此外,我们还证明了一个长度为 (ω left( frac{n}{dln d}right) )的循环的可能存在性。这些结果不仅概括了超立方体的情况,而且大大改进了超立方体的已知边界,特别是在(L_1)的可能直径和典型混合时间方面,以前只知道是 d 的多项式。
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引用次数: 0
Flashes and Rainbows in Tournaments 比赛中的闪光和彩虹
IF 1.1 2区 数学 Q1 MATHEMATICS Pub Date : 2024-04-04 DOI: 10.1007/s00493-024-00090-7
António Girão, Freddie Illingworth, Lukas Michel, Michael Savery, Alex Scott

Colour the edges of the complete graph with vertex set ({{1, 2, dotsc , n}}) with an arbitrary number of colours. What is the smallest integer f(lk) such that if (n > f(l,k)) then there must exist a monotone monochromatic path of length l or a monotone rainbow path of length k? Lefmann, Rödl, and Thomas conjectured in 1992 that (f(l, k) = l^{k - 1}) and proved this for (l ge (3 k)^{2 k}). We prove the conjecture for (l ge k^3 (log k)^{1 + o(1)}) and establish the general upper bound (f(l, k) le k (log k)^{1 + o(1)} cdot l^{k - 1}). This reduces the gap between the best lower and upper bounds from exponential to polynomial in k. We also generalise some of these results to the tournament setting.

用任意数量的颜色给顶点集 ({{1, 2, dotsc, n}}) 的完整图的边着色。如果 (n > f(l,k)) 那么一定存在长度为 l 的单调单色路径或长度为 k 的单调彩虹路径,那么 f(l, k) 的最小整数是多少?Lefmann、Rödl 和 Thomas 在 1992 年猜想 (f(l, k) = l^{k - 1} 并证明了 (l ge (3 k)^{2 k} 的这一猜想。)我们证明了 (lge k^3 (log k)^{1 + o(1)}) 的猜想,并建立了一般上界 (f(l, k) le k (log k)^{1 + o(1)}cdot l^{k - 1}).这将最佳下界和上界之间的差距从指数级缩小到 k 的多项式级。
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引用次数: 0
Lattice Path Matroids and Quotients 网格路径矩阵和阶乘
IF 1.1 2区 数学 Q1 MATHEMATICS Pub Date : 2024-04-04 DOI: 10.1007/s00493-024-00085-4
Carolina Benedetti-Velásquez, Kolja Knauer

We characterize the quotients among lattice path matroids (LPMs) in terms of their diagrams. This characterization allows us to show that ordering LPMs by quotients yields a graded poset, whose rank polynomial has the Narayana numbers as coefficients. Furthermore, we study full lattice path flag matroids and show that—contrary to arbitrary positroid flag matroids—they correspond to points in the nonnegative flag variety. At the basis of this result lies an identification of certain intervals of the strong Bruhat order with lattice path flag matroids. A recent conjecture of Mcalmon, Oh, and Xiang states a characterization of quotients of positroids. We use our results to prove this conjecture in the case of LPMs.

我们根据格子路径矩阵(LPM)的图来描述它们之间的商。通过这种描述,我们可以证明,用商数对 LPM 进行排序可以得到一个分级正集,它的秩多项式的系数是纳拉亚纳数。此外,我们还研究了全格路径旗状矩阵,并证明与任意正方体旗状矩阵相反,它们对应于非负旗变中的点。这一结果的基础是强布鲁特阶的某些区间与格状路径旗状矩阵的识别。麦卡蒙、吴和项最近的一个猜想指出了正交子的商的特征。我们用我们的结果证明了 LPM 的这一猜想。
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引用次数: 0
Boolean Function Analysis on High-Dimensional Expanders 高维扩展器上的布尔函数分析
IF 1.1 2区 数学 Q1 MATHEMATICS Pub Date : 2024-03-18 DOI: 10.1007/s00493-024-00084-5
Yotam Dikstein, Irit Dinur, Yuval Filmus, Prahladh Harsha

We initiate the study of Boolean function analysis on high-dimensional expanders. We give a random-walk based definition of high-dimensional expansion, which coincides with the earlier definition in terms of two-sided link expanders. Using this definition, we describe an analog of the Fourier expansion and the Fourier levels of the Boolean hypercube for simplicial complexes. Our analog is a decomposition into approximate eigenspaces of random walks associated with the simplicial complexes. Our random-walk definition and the decomposition have the additional advantage that they extend to the more general setting of posets, encompassing both high-dimensional expanders and the Grassmann poset, which appears in recent work on the unique games conjecture. We then use this decomposition to extend the Friedgut–Kalai–Naor theorem to high-dimensional expanders. Our results demonstrate that a constant-degree high-dimensional expander can sometimes serve as a sparse model for the Boolean slice or hypercube, and quite possibly additional results from Boolean function analysis can be carried over to this sparse model. Therefore, this model can be viewed as a derandomization of the Boolean slice, containing only (|X(k-1)|=O(n)) points in contrast to (left( {begin{array}{c}n kend{array}}right) ) points in the (k)-slice (which consists of all n-bit strings with exactly k ones).

我们开始研究高维扩展器的布尔函数分析。我们给出了基于随机漫步的高维扩展定义,这与早先用双面链接扩展器给出的定义不谋而合。利用这一定义,我们描述了简单复数的傅里叶展开和布尔超立方的傅里叶级数的类似方法。我们的类比是将与简单复数相关的随机漫步分解为近似的特征空间。我们的随机漫步定义和分解还有一个优势,即它们可以扩展到更一般的正集,包括高维扩展和格拉斯曼正集,这在最近关于唯一博弈猜想的研究中出现过。然后,我们利用这种分解将 Friedgut-Kalai-Naor 定理扩展到高维扩展集。我们的结果表明,常度高维扩展器有时可以作为布尔切片或超立方的稀疏模型,而且布尔函数分析的其他结果很有可能可以延续到这个稀疏模型中。因此,这个模型可以被看作是布尔切片的去随机化,只包含(|X(k-1)|=O(n))点,而不是(k)-切片(由所有 n 位字符串组成,其中正好有 k 个一)中的((left( {begin{array}{c}n kend{array}right) )点。)
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引用次数: 0
Ramsey Problems for Monotone Paths in Graphs and Hypergraphs 图和超图中单调路径的拉姆齐问题
IF 1.1 2区 数学 Q1 MATHEMATICS Pub Date : 2024-02-28 DOI: 10.1007/s00493-024-00082-7
Lior Gishboliner, Zhihan Jin, Benny Sudakov

The study of ordered Ramsey numbers of monotone paths for graphs and hypergraphs has a long history, going back to the celebrated work by Erdős and Szekeres in the early days of Ramsey theory. In this paper we obtain several results in this area, establishing two conjectures of Mubayi and Suk and improving bounds due to Balko, Cibulka, Král and Kynčl. For example, in the graph case, we show that the ordered Ramsey number for a fixed clique versus a fixed power of a monotone path of length n is always linear in n. Also, in the 3-graph case, we show that the ordered Ramsey number for a fixed clique versus a tight monotone path of length n is always polynomial in n. As a by-product, we also obtain a color-monotone version of the well-known Canonical Ramsey Theorem of Erdős and Rado, which could be of independent interest.

对图和超图的单调路径的有序拉姆齐数的研究由来已久,可以追溯到拉姆齐理论早期 Erdős 和 Szekeres 的著名工作。在本文中,我们获得了这一领域的若干结果,确立了 Mubayi 和 Suk 的两个猜想,并改进了 Balko、Cibulka、Král 和 Kynčl 的边界。例如,在图的情况下,我们证明了长度为 n 的固定簇与单调路径的固定幂的有序拉姆齐数总是与 n 成线性关系。此外,在 3 图的情况下,我们证明了长度为 n 的固定簇与紧密单调路径的有序拉姆齐数总是与 n 成多项式关系。作为副产品,我们还得到了厄多斯和拉多著名的 Canonical Ramsey Theorem 的彩色单调版本,这可能会引起人们的兴趣。
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引用次数: 0
The Ungar Games 温加尔游戏
IF 1.1 2区 数学 Q1 MATHEMATICS Pub Date : 2024-02-21 DOI: 10.1007/s00493-024-00083-6
Colin Defant, Noah Kravitz, Nathan Williams

Let L be a finite lattice. Inspired by Ungar’s solution to the famous slopes problem, we define an Ungar move to be an operation that sends an element (xin L) to the meet of ({x}cup T), where T is a subset of the set of elements covered by x. We introduce the following Ungar game. Starting at the top element of L, two players—Atniss and Eeta—take turns making nontrivial Ungar moves; the first player who cannot do so loses the game. Atniss plays first. We say L is an Atniss win (respectively, Eeta win) if Atniss (respectively, Eeta) has a winning strategy in the Ungar game on L. We first prove that the number of principal order ideals in the weak order on (S_n) that are Eeta wins is (O(0.95586^nn!)). We then consider a broad class of intervals in Young’s lattice that includes all principal order ideals, and we characterize the Eeta wins in this class; we deduce precise enumerative results concerning order ideals in rectangles and type-A root posets. We also characterize and enumerate principal order ideals in Tamari lattices that are Eeta wins. Finally, we conclude with some open problems and a short discussion of the computational complexity of Ungar games.

让 L 是一个有限网格。受到昂格尔对著名的斜坡问题的解答的启发,我们将昂格尔移动定义为将元素 (xin L) 发送到 ({x}cup T) 的满足的操作,其中 T 是 x 所覆盖的元素集合的子集。我们引入以下昂格尔博弈。从 L 的顶元素开始,两位棋手--阿特尼斯(Atniss)和埃塔(Eeta)--轮流走非难的昂格尔棋;谁先走不成昂格尔棋,谁就输掉对局。阿特尼斯先下。我们首先证明,在 (S_n) 的弱阶中,Eeta 赢的主阶理想数是 (O(0.95586^nn!))。然后,我们考虑了杨格中包括所有主阶理想的一大类区间,并描述了这一类区间中的 Eeta wins;我们推导出了关于矩形和 A 型根集合中阶理想的精确枚举结果。我们还表征并枚举了塔马里网格中属于埃塔胜的主阶理想。最后,我们以一些未决问题和对昂加博弈计算复杂性的简短讨论作结。
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引用次数: 0
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Combinatorica
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