Expansion in supercritical random subgraphs of the hypercube and its consequences
Joshua Erde, Mihyun Kang, Michael Krivelevich
备注：
29 pages, this work reuses parts from an earlier work of the same authors (arXiv:2106.04249, not to be published), we strengthen the main result of that work and give further applications
众所周知，当$p$在$\frac{1}{d}$附近时，$d$维超立方体的随机子图的行为经历相变，其中我们以概率$p$独立地包括每个边。更准确地说，标准论证表明，在$p$的这个值之下，这个图的所有分量都具有概率趋于1的顺序$O(d)$，如$d \to \infty$（简称whp），而Ajtai，Komlós和Szemerédi [$k$ -立方体的最大随机分量，Combinatorica 2（1982），no.1，1 - 7; MR 0671140]表明，刚好高于这个值，在超临界状态，whp有一个独特的“巨人”组成部分的顺序$Θ\left(2^d\right)$。证明了在$d$中巨分支的顶点展开是逆多项式。作为结果，我们获得了关于巨分支的直径和巨分支上的惰性随机游动的混合时间的在$d$界中的多项式，回答了Bollobás，Kohayakawa和Jumuczak的问题[On the diameter and radius of random subgraphs of the cube，Random Structures and Algorithms 5（1994），no.5，627- 648; MR 1300592]和Pete [A note on percolation on $\mathbb{Z}^d$：isoperimetric profile via exponential cluster repulsion，Electron. Commun.可能吧13（2008），377- 392; MR2415145]。此外，我们的结果意味着一个随机子图的超立方体的周长和Hadwiger数的下界在这个政权的$p$紧到多项式因子在$d$。
英文摘要：
It is well-known that the behaviour of a random subgraph of a $d$-dimensional hypercube, where we include each edge independently with probability $p$, undergoes a phase transition when $p$ is around $\frac{1}{d}$. More precisely, standard arguments show that just below this value of $p$ all components of this graph have order $O(d)$ with probability tending to one as $d \to \infty$ (whp for short), whereas Ajtai, Komlós and Szemerédi [Largest random component of a $k$-cube, Combinatorica 2 (1982), no. 1, 1--7; MR0671140] showed that just above this value, in the supercritical regime, whp there is a unique `giant' component of order $Θ\left(2^d\right)$. We show that whp the vertex-expansion of the giant component is inverse polynomial in $d$. As a consequence we obtain polynomial in $d$ bounds on the diameter of the giant component and the mixing time of the lazy random walk on the giant component, answering questions of Bollobás, Kohayakawa and Łuczak [On the diameter and radius of random subgraphs of the cube, Random Structures and Algorithms 5 (1994), no. 5, 627--648; MR1300592] and of Pete [A note on percolation on $\mathbb{Z}^d$: isoperimetric profile via exponential cluster repulsion, Electron. Commun. Probab. 13 (2008), 377--392; MR2415145]. Furthermore, our results imply lower bounds on the circumference and Hadwiger number of a random subgraph of the hypercube in this regime of $p$ which are tight up to polynomial factors in $d$.
Institute of Science and Technology Austria（奥地利科学技术研究所）；School of Mathematical Sciences, Tel Aviv University（特拉维夫大学理学院）；Institute of Discrete Mathematics, Graz University of Technology（格拉茨工业大学离散数学研究所）；Technical University of Vienna（维也纳工业大学）
A perfect matching reciprocity method for embedding multiple hypercubes in an augmented cube: Applications to Hamiltonian decomposition and fault-tolerant Hamiltonicity
The evolution of random subgraphs of the permutahedron
Instituto de Matemática, Estatística e Ciência da Computação, Universidade de São Paulo（圣保罗大学数学、统计和计算科学学院）；RobotDreams GmbH（RobotDreams有限公司）；School of Mathematics, University of Birmingham（伯明翰大学数学学院）
Department of Applied Mathematics, Faculty of Mathematics and Physics, Charles University（应用数学系，数学物理学院，查理大学）；Computer Science Institute, Faculty of Mathematics and Physics, Charles University（计算机科学研究所，数学物理学院，查理大学）
Tittmann，Averbouch and Makowsky [P. Tittmann，I. Averbouch，J.A. Makowsky，The enumeration of vertex induced subgraphs without the number of components，European Journal of Combinatorics，32（2011）954-974]，引入了子图分量多项式$Q(G;x,y)$，其对顶点诱导子图中的连通分量的数量进行计数。它包含了许多底层图的结构信息，例如顺序，大小，独立数。我们证明了正则二部图的连通度、长度为4的圈数等图的不变量是由子图的分量多项式决定的。利用所得到的结果，我们发现了几类著名的图是由$Q(G;x,y)$确定的：路、圈、蝌蚪图、完全二部图、友谊图、书图和超立方体.此外，我们还研究了图的区分能力，发现了一些不能用子图分量多项式区分而只能用Tutte多项式和特征多项式区分的单图。这是对Tittmann等人提出的关于子图分量多项式的三个问题的回答。
本文研究了由$k$ -out模型生成的$n$ -立方体的随机子图的连通性，记为$Q^n(k)$。令$k$为整数$1\leq k \leq n-1$。我们让$Q^n(k)$是通过独立地包括每个$v\in V(Q^n)$的一组$k$不同的边缘，从所有的$\binom{n}{k}$集的不同的边缘，是事件$v$均匀选择生成的图。我们研究了当$k$变化时，$Q^n(k)$的连通性.我们发现，w. h. p. $Q^n(1)$不包含一个巨大的组成部分，即一个组件，跨越$Ω(2^n)$顶点。此后，我们表明，这样的组件出现时$k=2$。此外，巨分量跨越除了$o(2^n)$之外的所有顶点，因此它是唯一的。然后，我们建立在$k_0= \log_2 n -2\log_2\log_2 n $上找到的连接阈值。阈值是尖锐的意义上说，$Q^n(\lfloor k_0\rfloor )$是断开的，但$Q^n(\lceil k_0\rceil+1)$连接w. h. p.此外，我们表明，w. h. p. $Q^n(k)$是$k$ -连接的每一个$k\geq \lceil k_0\rceil+1$。
