Hamilton completion and the path cover number of sparse random graphs

We prove that for every epsilon>0 there is c0 such that if G similar to G(n,c/n), c >= c(0), then with high probability G can be covered by at most (1+epsilon)& sdot;(1)/(2)ce(-c)& sdot;n vertex disjoint paths, which is essentially tight. This is equivalent to showing that, with high probability, at most (1+epsilon)& sdot;(1)/(2)ce(-c)& sdot;n edges can be added to G to create a Hamiltonian graph.