For a graph G, p(G) denotes the order of a longest path in G and c(G) the order of a longest cycle. We show that if G is a connected graph n ≥ 3 vertices such that d(u) + d(v) + d(w) n for all triples u, v, w of independent vertices, then G satisfies c(G) ≥ p(G) - 1, or G is in one of six families of exceptional graphs. This generalizes results of Bondy and of Bauer, Morgana, Schmeichel, and Veldman.
Enomoto, H., van den Heuvel, J., Kaneko, A., & Saito, A. (1995). Relative length of long paths and cycles in graphs with large degree sums. Journal of graph theory, 201(2), 213-225. https://doi.org/10.1002/jgt.3190200210