On matchings and hamiltonian cycles in random graphs Annals of Discrete Mathematics 28

Let m = ¼ n log n + ½ n log log n + c n . Let Λ denote the set of graphs with vertices {1, 2, …, n }, m edges and minimum degree 1. We show that if a random graph G is chosen uniformly from Λ then We also show that if a random graph G with vertices {1, 2, …, n } is constructed by randomly adding edges one at a time then, almost surely, as soon as G has degree k, G has [ k /2] disjoint hamiltonian cycles plus a disjoint perfect matching if k is odd, where k is a fixed positive integer.

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