A note on 2-factors with two components

In this note, we consider a minimum degree condition for a hamiltonian graph to have a 2-factor with two components. Let G be a graph of order n⩾3. Dirac's theorem says that if the minimum degree of G is at least 12n, then G has a hamiltonian cycle. Furthermore, Brandt et al. [J. Graph Theory 24 (1997) 165-173] proved that if n⩾8, then G has a 2-factor with two components. Both theorems are sharp and there are infinitely many graphs G of odd order and minimum degree 12(|G|-1) which have no 2-factor. However, if hamiltonicity is assumed, we can relax the minimum degree condition for the existence of a 2-factor with two components. We prove in this note that a hamiltonian graph of order n⩾6 and minimum degree at least 512n+2 has a 2-factor with two components.