A number of results are established concerning long cycles in graphs with large degree sums. Let G be a graph on n vertices such that d(x) + d(y) + d(z) 3s for all triples of independent vertices x, y, z. Let c be the length of a longest cycle in G and (Y the cardinality of a maximum independent set of vertices. If G is l-tough and s an, then every longest cycle in G is a dominating cycle and c z min(n, n + fs-cu) >, min(n , $n + 4s) 3 &a. If G is 2-connected and s 2 n + 2, then also c 3 min(n, n + 4s-(u), generalizing a result of Bondy and one of Nash-Williams. Finally, if G is 2-tough and s 2 n, then G is hamiltonian.

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