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(v)+d(w)~> n + 2 for every triple u, v, w of independent vertices. As a corollary we obtain the following improvement of a conjecture of H/iggkvist (1992): Let G be a 2-connected graph on n vertices where every pair of nonadjacent vertices has degree sum at least n-k and assume G has a 2-factor with at least k+ 1 odd components. Then G is hamiltonian. 1. Results We use [4] for terminology and notation not defined here and consider finite, simple graphs only. The following three conjectures, among many others, appear in [6]. Conjecture 1. Let G be a 2-connected graph on n vertices where every pair of nonadjacent vertices has degree sum at least n-k and assume furthermore that G has a k-factor. Then G is hamiltonian. Conjecture 2. Let G be a 2-connected graph on n vertices where every pair of nonadjacent vertices has degree sum at least n-k and assume that G has a 2-factor where every component is of order more than k. Then G is hamiltonian. Conjecture 3. Let G be a 2-connected graph on n vertices where every pair of nonadjacent vertices has degree sum at least n-k and assume that G has a 2-factor with at least 2k odd components. Then G is hamiltonian.