On the trace of random walks on random graphs

We study graph-theoretic properties of the trace of a random walk on a random graph. We show that for any epsilon>0 there exists C>1 such that the trace of the simple random walk of length (1+epsilon)nlnn on the random graph G approximate to G(n,p) for p>Clnn/n is, with high probability, Hamiltonian and (lnn)-connected. In the special case p=1 (that is, when G=Kn), we show a hitting time result according to which, with high probability, exactly one step after the last vertex has been visited, the trace becomes Hamiltonian, and one step after the last vertex has been visited for the k'th time, the trace becomes 2k-connected.