Long cycles in graphs containing a 2-factor with many odd components

We prove a result on the length of a longest cycle in a graph on n vertices that contains a 2-factor and satisfies d(u)+d(c)+d(w)⩾n+2 for every tiple u, v, w of independent vertices. As a corollary we obtain the follwing improvement of a conjectre of (1992): Let G be a 2-connected graph on n vertices where every pair of nonadjacent vertices has degree sum at least −k and assume G has a 2-factor with at least k+1 odd components. Then G is hamiltonian.