Hamilton cycles in random lifts of graphs

An n-lift of a graph K is a graph with vertex set V(K)×n], and for each edge (i,j)E(K) there is a perfect matching between {i}×n] and {j}×n]. If these matchings are chosen independently and uniformly at random then we say that we have a random n-lift. We show that there are constants h₁,h₂ such that if h≥h₁ then a random n-lift of the complete graph K_h is hamiltonian and if h≥h₂ then a random n-lift of the complete bipartite graph K_h,h is hamiltonian .