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首页> 外文期刊>Journal of Combinatorial Theory, Series B >On a perfect matching in a random digraph with average out-degree below two (Mar, 10.1016/S0095-8956(03)00024-8, 2020)
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On a perfect matching in a random digraph with average out-degree below two (Mar, 10.1016/S0095-8956(03)00024-8, 2020)

机译：在随机性数字中的完美匹配，平均出距离2（Mar，10.1016 / S0095-8956（03）00024-8,2020）

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Existence of a perfect matching in a random bipartite digraph with bipartition (V-1, V-2), vertical bar V-i vertical bar = 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 mvertices 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 + P(Poisson(1) <= m) is an element of(1, 2). Aided by Matlab software, we prove that for m = 1, whence for all m >= 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 nonbipartite version of this model with vertex set [n] we show that already for m = 0 a.a.s. there exists a partialmatching which leaves unmatched a O(log-1n) fraction of vertices. (C) 2020 Elsevier Inc. All rights reserved.
机译：研究了在随机二群数字中的完美匹配，具有两分之一（V-1，V-2），垂直条V-I垂直条= n。该图是在潜在的匹配伙伴的两轮随机选择中生成的，使得每个顶点的平均选择数量总体在2中低于2。更确切地说，在第一轮中，每个顶点在随机均匀地选择潜在的伴侣，并且独立地选择潜在的伴侣所有顶点。如果固定的整数M，则Vertex被分类为不受欢迎，如果它由来自另一侧的大多数射门选择。每个不受欢迎的顶点都制作了另一个均匀/独立的潜在伴侣的选择。通用顶点V，即其Out度，渐近至1 + P（泊松（1）<= M）是（1,2）的元素的预期选择。由MATLAB软件辅助，我们证明了对于M = 1，对所有M> = 1的作用，所得到的二分图具有完美的匹配A.。 （渐近几乎肯定地）。另一方面，对于m = 0 a.a.s.完美的匹配不存在。对于具有顶点集的本型号的非编排版本[n]，我们向m = 0 a.a.s显示已经显示。存在偏离的部分，其留下了无与伦比的顶点的O（log-1n）分数。 （c）2020 Elsevier Inc.保留所有权利。

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