The author proves 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) = or > n + 2 for every triple u, v, w of independent vertices. As a corollary, the author obtains the following improvement of a conjecture of R. Haggkvist: Let G be a 2-connected graph on n vertices where every pair of nonadjacent vertices has degree sum at least max (n - k, 2/3n + 1) and assume G has a 2-factor with at least k - 1 odd components. Then G is hamiltonian.
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