Sandwiching biregular random graphs

Abstract

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 $(n \log n)^{3/4}$.