Hamilton cycles in random subgraphs of pseudo-random graphs

Given an r-regular graph G on n vertices with a Hamilton cycle, order its edges randomly and insert them one by one according to the chosen order, starting from the empty graph. We prove that if the eigenvalue of the adjacency matrix of G with the second largest absolute value satisfies λ = o(r^5/2/(n^3/2(log n)^3/2)), then for almost all orderings of the edges of G at the very moment τ* when all degrees of the obtained random subgraph H_t* of G become at least two, H_τ* has a Hamilton cycle. As a consequence we derive the value of the threshold for the appearance of a Hamilton cycle in a random subgraph of a pseudo-random graph G, satisfying the above stated condition.