Disjoint hamiltonian cycles in bipartite graphs

Abstract

Let G = ( X , Y ) be a bipartite graph and define σ 2 2 ( G ) = min { d ( x ) + d ( y ) : x y ∉ E ( G ) , x ∈ X , y ∈ Y } . Moon and Moser [J. Moon, L. Moser, On Hamiltonian bipartite graphs, Israel J. Math. 1 (1963) 163–165. MR 28 # 4540] showed that if G is a bipartite graph on 2 n vertices such that σ 2 2 ( G ) ≥ n + 1 , then G is hamiltonian, sharpening a classical result of Ore [O. Ore, A note on Hamilton circuits, Amer. Math. Monthly 67 (1960) 55] for bipartite graphs. Here we prove that if G is a bipartite graph on 2 n vertices such that σ 2 2 ( G ) ≥ n + 2 k − 1 , then G contains k edge-disjoint hamiltonian cycles. This extends the result of Moon and Moser and a result of R. Faudree et al. [R. Faudree, C. Rousseau, R. Schelp, Edge-disjoint Hamiltonian cycles, Graph Theory Appl. Algorithms Comput. Sci. (1984) 231–249].