A common generalization of Chvátal-Erdős' and Fraisse's sufficient conditions for hamiltonian graphs

Abstract

Let G = ( V, E) be a k-connected graph of order n. For S ⊂ V, let N( S) be its neighborhood set and let J( S) = { u ∉ S | N( u) ⊇ S} if | S| ⩾ 2 and J( S) = 0 otherwise. If there exists some s, 1 ⩽ s ⩽ k, such that every independent set X of s + 1 vertices has a vertex u satisfying | N( Xβ{ u})| + | N( u) ∪ J( X) β{ u})| ⩾ n, then G is hamiltonian. From this main theorem, we derive new sufficient conditions for hamiltonian graphs. Some of these results are improvements and/or generalizations of various known results. In particular, sufficient conditions of Ore (1960), Chvátal and Erdős (1972), Fraisse (1986) and E. Flandrin et al. (1991) are easily derived.