Sandwiching biregular random graphs

Abstract Let ${\mathbb{G}(n_1,n_2,m)}$ be a uniformly random m -edge subgraph of the complete bipartite graph ${K_{n_1,n_2}}$ with bipartition $(V_1, V_2)$ , where $n_i = |V_i|$ $i=1,2$ . Given real number $p \in [0,1]$ such that $d_1 \,{:\!=}\, pn_2$ and $d_2 pn_1$ are integers, let $\mathbb{R}(n_1,n_2,p)$ every vertex $v V_i$ degree $d_i$ $i 1, 2$ In this paper we determine sufficient conditions on $n_1,n_2,p$ under which one can embed into vice versa probability tending to 1. particular, in balanced case $n_1=n_2$ show if $p\gg\log n/n$ $1 - p \gg \left(\log n/n \right)^{1/4}$ then for some $m\sim pn^2$ asymptotically almost surely while $p\gg\left(\log^{3} n/n\right)^{1/4}$ $1-p\gg\log opposite embedding holds. As an extension, confirm Kim–Vu Sandwich Conjecture degrees growing faster than $(n \log n)^{3/4}$