A note on 2-factors with two components

In this talk, 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 n/2, then G has a hamiltonian cycle. Furthermore, Brandt et al. 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 (|G| − 1)/2 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. A hamiltonian graph of order n ≥ 6 and minimum degree 5n/12+ 2 has a 2-factor with two components.