Graphs and Combinatorics (2000) 16 : 67±80 Graphs and Combinatorics ( Springer-Verlag 2000 Cycles in 2-Factors of Balanced Bipartite Graphs Guantao Chen1*, Ralph J. Faudree 2y , Ronald J. Gould 3z , Michael S. Jacobson4 § , and Linda Lesniak5 P 1 Georgia State University, Atlanta, GA 30303, USA 2 University of Memphis, Memphis, TN 38152, USA 3 Emory University, Atlanta GA 30322, USA 4 University of Louisville, Louisville, KY 40292, USA 5 Drew University, Madison NJ 07940, USA Abstract. In the study of hamiltonian graphs, many well known results use degree con- ditions to ensure su½cient edge density for the existence of a hamiltonian cycle. Recently it was shown that the classic degree conditions of Dirac and Ore actually imply far more than the existence of a hamiltonian cycle in a graph G, but also the existence of a 2-factor with exactly k cycles, where 1 U k U jV Gj 4 . In this paper we continue to study the number of cycles in 2-factors. Here we consider the well-known result of Moon and Moser which implies the existence of a hamiltonian cycle in a balanced bipartite graph of order 2n. We show that a related degree condition also implies the existence of a 2-factor with exactly k cycles in a balanced bipartite graph of order 2n with n V max 51; k 2 2  1   . 1. Introduction All graphs considered are simple, without loops or multiple edges. A 2-factor of a graph G is a 2-regular subgraph of G that spans the vertex set V G, that is, a 2-factor is a collection of vertex disjoint cycles that cover all vertices of G. For years mathematicians have investigated results ensuring the existence of 2-factors in graphs. Hundreds of results exist concerning the special case when the graph is hamiltonian, that is, the 2-factor is a single cycle. Recently, there have been e¨orts to determine more about the structure of general 2-factors. Questions about the number of cycles possible in a 2-factor or the lengths of the cycles forming the 2-factor have drawn interest. * Supported by N.S.A. Grant MDA904-97-1-0101 y Supported by O.N.R. Grant N00014-91-J-1085 z Supported by O.N.R. Grant N00014-97-1-0499 § Supported by O.N.R. Grant N00014-91-J-1098 P Supported by O.N.R. Grant N00014-J-93-1-0050