On a perfect matching in a random bipartite digraph with average out-degree below two

Existence of a perfect matching in a random bipartite digraph with bipartition $(V_1, V_2)$, $|V_i|=n$, is studied. The graph is generated in two rounds of random selections of a potential matching partner such that the average number of selections made by each vertex overall is below $2$. More precisely, in the first round each vertex chooses a potential mate uniformly at random, and independently of all vertices. Given a fixed integer $m$, a vertex is classified as unpopular if it has been chosen by at most $m$ vertices from the other side. Each unpopular vertex makes yet another uniform/independent selection of a potential mate. The expected number of selections made by a generic vertex $v$, i.e. its out-degree, is asymptotic to $1+\Bbb P(\text{Poisson}(1)\le m)\in (1,2)$. Aided by Matlab software, we prove that for $m=1$, whence for all $m\ge 1$, the resulting bipartite graph has a perfect matching a.a.s. (asymptotically almost surely). On the other hand, for $m=0$ a.a.s. a perfect matching does not exist, and the graph consists of a single giant component of size $2n -O(n^{1/2+o(1)})$ and possibly some components of size $O(\log n)$. This is a thorough revision of the joint paper (JCT(B) 88 (2003), 1-16) by the first author and the third author.