Sandwiching Biregular Random Graphs

Abstrakt

Let G(n1,n2,m) be a uniformly random m-edge subgraph of the complete bipartite graph Kn1,n2 with bipartition (V1,V2) , where ni=|Vi| , i=1,2 . Given a real number p∈[0,1] such that d1:=pn2 and d2:=pn1 are integers, let R(n1,n2,p) be a random subgraph of Kn1,n2 with every vertex v∈Vi of degree di , i=1,2 . In this paper we determine sufficient conditions on n1,n2,p and m under which one can embed G(n1,n2,m) into R(n1,n2,p) and vice versa with probability tending to 1. In particular, in the balanced case n1=n2 , we show that if p≫logn/n and 1−p≫(logn/n)1/4 , then for some m∼pn2 , asymptotically almost surely one can embed G(n1,n2,m) into R(n1,n2,p) , while for p≫(log3n/n)1/4 and 1−p≫logn/n the opposite embedding holds. As an extension, we confirm the Kim–Vu Sandwich Conjecture for degrees growing faster than (nlogn)3/4 .