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On a conjecture of Conlon, Fox, and Wigderson 关于康伦、福克斯和维格德森的一个猜想

Combinatorics, Probability and Computing

Pub Date : 2024-02-16 DOI: 10.1017/s0963548324000026

Chunchao Fan, Qizhong Lin, Yuanhui Yan

For graphs

$G$

and

$H$

, the Ramsey number

$r(G,H)$

is the smallest positive integer

$N$

such that any red/blue edge colouring of the complete graph

$K_N$

contains either a red

$G$

or a blue

$H$

. A book

$B_n$

is a graph consisting of

$n$

triangles all sharing a common edge. Recently, Conlon, Fox, and Wigderson conjectured that for any

$0lt alpha lt 1$

, the random lower bound

对于图形 $G$ 和 $H$，拉姆齐数 $r(G,H)$ 是最小的正整数 $N$，使得完整图形 $K_N$ 的任何红/蓝边着色都包含红色 $G$ 或蓝色 $H$。一本书 $B_n$ 是由所有共享一条公共边的 $n$ 三角形组成的图形。最近，Conlon、Fox 和 Wigderson 猜想，对于任意 $0lt alpha lt 1$，随机下界 $r(B_{lceil alpha nrceil },B_n)ge (sqrt{alpha }+1)^2n+o(n)$ 并不严密。换句话说，存在某个常数 $beta gt (sqrt{alpha }+1)^2$ ，使得 $r(B_{lceil alpha nrceil },B_n)ge beta n$ 对于所有足够大的 $n$ 。根据 Nikiforov 和 Rousseau 2005 年的一个结果，这个猜想在每一个 $alpha lt 1/6$ 都成立，这个结果说在这个范围内，对于所有足够大的 $n$ ，$r(B_{lceil alpha nrceil },B_n)=2n+3$ 。我们推翻了康伦、福克斯和维格德森的猜想。事实上，我们证明了随机下界对于每 1/4 美元（α）和 1 美元（leq 1）都是渐近紧密的。此外，我们还证明了对于任意1/6leq alpha le 1/4$和大$n$，$r(B_{lceil alpha nrceil }, B_n)le left (frac 32+3alpha right ) n+o(n)$ ，其中当$alpha =1/6$或1/4$时，不等式是渐近紧密的。我们还给出了 $r(B{lceil alpha nrceil }, B_n)$ 为 $1/6le alpha lt frac{52-16sqrt{3}}{121}approx 0.2007$ 的下界，表明随机下界并不严密，即 Conlon、Fox 和 Wigderson 的猜想在此区间内成立。

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引用次数: 0

Percolation on irregular high-dimensional product graphs 不规则高维积图上的循环

Combinatorics, Probability and Computing

Pub Date : 2023-12-20 DOI: 10.1017/s0963548323000469

Sahar Diskin, Joshua Erde, Mihyun Kang, Michael Krivelevich

We consider bond percolation on high-dimensional product graphs $G=square _{i=1}^tG^{(i)}$, where $square$ denotes the Cartesian product. We call the $G^{(i)}$ the base graphs and the product graph $G$ the host graph. Very recently, Lichev (J. Graph Theory, 99(4):651–670, 2022) showed that, under a mild requirement on the isoperimetric properties of the base graphs, the component structure of the percolated graph $G_p$ undergoes a phase transition when $p$ is around $frac{1}{d}$, where $d$ is the average degree of the host graph.

In the supercritical regime, we strengthen Lichev’s result by showing that the giant component is in fact unique, with all other components of order

我们考虑高维积图 $G=square _{i=1}^tG^{(i)}$ 上的键渗，其中 $square$ 表示笛卡尔积。我们称 $G^{(i)}$ 为基图，称积图 $G$ 为主图。最近，Lichev（J. Graph Theory, 99(4):651-670, 2022）指出，在对基图的等周特性有温和要求的情况下，当 $p$ 约为 $frac{1}{d}$（其中 $d$ 是主图的平均度数）时，渗滤图 $G_p$ 的分量结构会发生相变。在超临界体系中，我们证明了巨分量实际上是唯一的，所有其他分量的阶数都是 $o(|G|)$，并确定了巨分量的尖锐渐近阶数，从而加强了利切夫的结果。此外，我们还回答了利切夫提出的两个问题（《图论》，99(4):651-670, 2022 年）：首先，我们提供了一个构造，表明有界度要求对于线性阶成分的可能出现是必要的；其次，我们证明了对基图的等度数要求实际上在维数上是超指数小的。最后，在次临界机制中，我们举例说明，在不规则高维积图的情况下，可以高概率地出现一个多项式大的分量，这与厄尔多斯-雷尼随机图和渗滤超立方体中的定量行为非常不同，事实上也与任何规则高维积图中的定量行为非常不同，作者在另一篇论文（Percolation on high-dimensional product graphs. arXiv:2209.03722, 2022）中也证明了这一点。

引用次数: 0

Product structure of graph classes with bounded treewidth 有界树宽图类的乘积结构

Combinatorics, Probability and Computing

Pub Date : 2023-12-07 DOI: 10.1017/s0963548323000457

Rutger Campbell, Katie Clinch, Marc Distel, J. Pascal Gollin, Kevin Hendrey, Robert Hickingbotham, Tony Huynh, Freddie Illingworth, Youri Tamitegama, Jane Tan, David R. Wood

We show that many graphs with bounded treewidth can be described as subgraphs of the strong product of a graph with smaller treewidth and a bounded-size complete graph. To this end, define the underlying treewidth of a graph class $mathcal{G}$ to be the minimum non-negative integer $c$ such that, for some function $f$, for every graph $G in mathcal{G}$ there is a graph $H$ with $textrm{tw}(H) leqslant c$ such that $G$ is isomorphic to a subgraph of $H boxtimes K_{f(textrm{tw}(G))}$. We introduce disjointed coverings of graphs and show they determine the underlying treewidth of any graph class. Using this result, we prove that the class of planar graphs has underlying treewidth

我们的研究表明，许多具有有界树宽（treewidth）的图可以被描述为具有较小树宽的图与有界大小的完整图的强积的子图。为此，我们将图类 $mathcal{G}$ 的底层树宽定义为最小非负整数 $c$，使得对于某个函数 $f$、对于每一个在 $mathcal{G}$ 中的图 $G，都有一个具有 $textrm{tw}(H) （leqslant c）的图 $H$，使得 $G$ 与 $H 的一个子图（boxtimes K_{f(textrm{tw}(G)）}$ 同构。我们引入了图的无接缝覆盖，并证明它们决定了任何图类的底层树宽。利用这一结果，我们证明了平面图类的底层树宽为 $3$；$K_{s,t}$-minor-free 图类的底层树宽为 $s$（对于 $t geqslant max {s,3}$）；$K_t$-minor-free 图类的底层树宽为 $t-2$。一般来说，我们证明了当且仅当一个单调类排除了某个固定的拓扑次要图时，它才具有有界的底层树宽。我们还研究了由排除子图或排除诱导子图定义的图类的底层树宽。我们证明，当且仅当 $H$ 的每个分量都是细分星形时，没有 $H$ 子图的图类具有有界底层树宽；当且仅当 $H$ 的每个分量都是星形时，没有诱导 $H$ 子图的图类具有有界底层树宽。

引用次数: 0

On minimum spanning trees for random Euclidean bipartite graphs 随机欧几里得二部图的最小生成树

Combinatorics, Probability and Computing

Pub Date : 2023-11-30 DOI: 10.1017/s0963548323000445

Mario Correddu, Dario Trevisan

We consider the minimum spanning tree problem on a weighted complete bipartite graph

$K_{n_R, n_B}$

whose

$n=n_R+n_B$

vertices are random, i.i.d. uniformly distributed points in the unit cube in

$d$

dimensions and edge weights are the

$p$

-th power of their Euclidean distance, with

$pgt 0$

. In the large

$n$

limit with

$n_R/n to alpha _R$

and

$0lt alpha _Rlt 1$

, we show that the maximum vertex degree of the tree grows logarithmically, in contrast with the classical, non-bipartite, case, where a uniform bound holds depending on

$d$

only. Despite this difference, for

$plt d$

考虑一个加权完全二部图$K_{n_R, n_B}$上的最小生成树问题，其中$n=n_R+n_B$顶点是随机的，$d$维的单位立方体中有i个均匀分布的点，边权为其欧几里得距离的$p$ -次幂，$pgt 0$。在具有$n_R/n 到$ α _R$和$0lt α _Rlt $ 1$的大$n$极限中，我们证明了树的最大顶点度是对数增长的，与经典的，非二部的情况相反，在这种情况下，一致界只依赖于$d$。尽管存在这种差异，对于plt d$，我们能够证明以速率$n^{1-p/d}$归一化的总边代价收敛于一个可以表示为一系列积分的极限常数，从而将Avram和Bertsimas的经典结果推广到二部情况，并证实了Riva, Caracciolo和Malatesta的一个猜想。

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引用次数: 0