Relative length of long paths and cycles in graphs with large degree sums

Abstract

Abstract 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. © 1995, John Wiley & Sons, Inc.