Maximum matchings in a class of random graphs
Journal of Combinatorial Theory Series B
Average case analysis of a heuristic for the assignment problem
Mathematics of Operations Research
The random bipartite nearest neighbor graphs
Random Structures & Algorithms - Special issue on statistical physics methods in discrete probability, combinatorics, and theoretical computer science
On Perfect Matchings and Hamilton Cycles in Sums of Random Trees
SIAM Journal on Discrete Mathematics
Karp-sipser on random graphs with a fixed degree sequence
Combinatorics, Probability and Computing
Maximum matchings in random bipartite graphs and the space utilization of Cuckoo Hash tables
Random Structures & Algorithms
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Structural properties of a random bipartite graph with bipartition (V1, V2), (|V1| = |V2|) = n), are studied. The graph is generated via two rounds of potential mates selections. In the first round every vertex in Vi chooses uniformly at random a vertex from Vj, j ≠ i, i = 1,2. In the second round each of the "unpopular" vertices, i.e. neglected completely in the first round, is allowed to make another random selection of a vertex. The resulting graph is "sandwiched" between Bn(1), the first-round graph, and Bn(2), the graph obtained by allowing every vertex to make two random selections of a mate. It seems natural to denote our graph Bn(1 + e-1), as the expected number of selections per vertex is 1 + e-1 in the limit. We prove that, asymptotically almost surely (a.a.s.), Bn(1 + e-1) contains a perfect matching, thus strengthening a well-known Walkup's theorem on a.a.s, existence of a perfect matching in the graph Bn(2). We demonstrate also that a.a.s. Bn(1 + e-1) consists of a giant component and several one-cycle components of a total size bounded in probability, and that Bn(1 + e-1) is connected with the limiting probability √1 - exp(-2(1 + e-1)) = 0.96703... perfect.