On 2-factors of a bipartite graph

In this article, we consider the following problem: Given a bipartite graph G and a positive integer k, when does G have a 2-factor with exactly k components? We will prove that if G = (V1, V2, E) is a bipartite graph with |V1| = |V2| = n ≥ 2k + 1 and δ (G) ≥ ⌈n/2⌉ + 1, then G contains a 2-factor with exactly k components. We conjecture that if G = (V1, V2; E) is a bipartite graph such that |V1| = |V2| = n ≥ 2 and δ (G) ≥ ⌈n/2⌉ + 1, then, for any bipartite graph H = (U1, U2; F) with |U1| ≤ n, |U2| ≤ n and Δ (H) ≤ 2, G contains a subgraph isomorphic to H. © 1999 John Wiley & Sons, Inc. J Graph Theory 31: 101–106, 1999