Cycles containing all vertices of maximum degree

For a graph Gand an integer k, denote by Vk the set {u E V(G) I d(u) 2 k}. Veldman proved that if G is a 2-connected graph of order n with n 5 3k 2 and IVkl 5 k, then G has a cycle containing all vertices of Vk. It is shown that the upper bound k on IVkl is close to best possible in general. For the special case k = A(G), it is conjectured that the condition lVkl I k can be omitted. Using a variation of Woodall's Hopping Lemma, the conjecture is proved under the additional condition that n 5 2A(G) + S(G) + 7. This result is an almost-generalization of Jackson's Theorem that every 2-connected k-regular graph of order n with n 5 3k is hamiltonian. An alternative proof of an extension of Jackson's Theorem is also presented.