Disjoint hamiltonian cycles in bipartite graphs

Let G = (X, Y ) be a bipartite graph and define σ 2(G) = min{d(x) + d(y) : xy / ∈ E(G), x ∈ X, y ∈ Y }. Moon and Moser [5] showed that if G is a bipartite graph on 2n vertices such that σ 2(G) ≥ n + 1 then G is hamiltonian, sharpening a classical result of Ore [6] for bipartite graphs. Here we prove that if G is a bipartite graph on 2n vertices such that σ 2(G) ≥ n+ 2k− 1 then G contains k edge-disjoint hamiltonain cycles. This extends the result of Moon and Moser and a result of R. Faudree, et al. [3]