夹层双正则随机图
Sandwiching biregular random graphs
摘要：
设$G(n,n,m)$是具有二分图$(V_1, V_2)$的完全二部图$K_{n,n}$的一致随机$m$ -边子图，其中$n_i = |V_i|$ .给定一个实数$p \in [0,1]$，使得$d_1 := pn_2$和$d_2 := pn_1$是整数，设$R(n,n,p)$是$K_{n,n}$的一个随机子图，使得每个$v \in V_i$都有度$d_i$，对于$i = 1, 2$。在本文中，我们确定充分条件$n_1,n_2,p$，$m$下，人们可以嵌入$G(n,n,m)$到$R(n,n,p)$，反之亦然，概率趋于$1$。特别是，在平衡的情况下$n_1 = n_2$，我们表明，如果$p \gg \log n/n$和$1 - p \gg \left(\log n/n \right)^{1/4}$，那么对于一些$m \sim pn^2$，渐近几乎肯定可以嵌入$G(n,n,m)$到$R(n,n,p)$，而对于$p \gg \left(\log^{3} n/n\right)^{1/4}$和$1-p \gg \log n/n$，我们有相反的嵌入。作为推广，我们证实了度增长快于$(n \log n)^{3/4}$的Kim-Vu三明治猜想.
英文摘要：
Let $G(n,n,m)$ be a uniformly random $m$-edge subgraph of the complete bipartite graph $K_{n,n}$ with bipartition $(V_1, V_2)$, where $n_i = |V_i|$. Given a real number $p \in [0,1]$ such that $d_1 := pn_2$ and $d_2 := pn_1$ are integers, let $R(n,n,p)$ be a random subgraph of $K_{n,n}$ such that every $v \in V_i$ has degree $d_i$, for $i = 1, 2$. In this paper we determine sufficient conditions on $n_1,n_2,p$, and $m$ under which one can embed $G(n,n,m)$ into $R(n,n,p)$ and vice versa with probability tending to $1$. In particular, in the balanced case $n_1 = n_2$, we show that if $p \gg \log n/n$ and $1 - p \gg \left(\log n/n \right)^{1/4}$, then for some $m \sim pn^2$, asymptotically almost surely one can embed $G(n,n,m)$ into $R(n,n,p)$, while for $p \gg \left(\log^{3} n/n\right)^{1/4}$ and $1-p \gg \log n/n$ we have the opposite embedding. As an extension, we confirm the Kim--Vu Sandwich Conjecture for degrees growing faster than $(n \log n)^{3/4}$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
