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 $1+\mathbb{P}(\text{Poisson}(1)\le m)\in (1,2)$. 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. For the non-bipartite version of this model with vertex set $[n]$ we show that already for $m=0$ a.a.s. there exists a partial matching which leaves unmatched a $O({\log }^{-1}n)$ fraction of vertices.

具有二等分的随机二部有向图的完美匹配的存在 $（V_{\mathrm{1个}}，V_2）$， $|V_{\mathrm{一世}}|=ñ$，正在研究中。该图是在两轮潜在匹配对象的随机选择中生成的，因此每个顶点总体选择的平均数目低于2。更准确地说，在第一轮中，每个顶点均会随机且独立地均匀选择一个潜在伴侣。所有顶点。给定一个固定的整数m，如果一个顶点最多被另一侧的m个顶点选中，则将其分类为不受欢迎。每个不受欢迎的顶点都会对潜在配偶进行另一个统一/独立的选择。通用顶点v的预期选择次数（即其出学位程度）渐近于$\mathrm{1个}+\mathbb{P}（\text{泊松}（\mathrm{1个}）\le 米）\in （\mathrm{1个}，2）$。在Matlab软件的帮助下，我们证明了$米=\mathrm{1个}$，所有人 $米\ge \mathrm{1个}$，生成的二部图具有完美的匹配aas（渐近几乎确定地）。另一方面，对于$米=0$aas不存在完美匹配。对于具有顶点集的该模型的非双向版本$[ñ]$ 我们已经证明了 $米=0$aas存在部分匹配项，从而留下了不匹配的a$Ø（{\mathrm{日志}}^{-\mathrm{1个}}ñ）$ 顶点的分数。