A generalization of Bondy's and Fan's sufficient conditions for Hamiltonian graphs

Abstract

Let G be a k-connected ( k ⩾ 2) graph on n vertices. Let S be an independent set of G. S is called essential if there exists two distinct vertices in S which have a common neighbor in G. In this paper we shall prove that if max { d( u) : u ∈ S} ⩾ n/2 holds for any essential independent set S with k + 1 vertices of G, then either G is hamiltonian or G is one of three classes of exceptional graphs. This is motivated by a result of Chen et al. (1994) and generalizes the results of Bondy (1980) and Fan (1984).