On the Hamiltonian-Connectedness for Graphs Satisfying Ore’s Theorem

Abstract

Consider any undirected and simple graph G = (V,E), where V and E denote the vertex set and the edge set of G, respectively. Let |G| = |V| = n ≥ 3. The well-known Ore’s theorem states that if deg _G (u) + deg _G (v) ≥ n holds for each pair of nonadjacent vertices u and v of G, then G is hamiltonian. A similar theorem given by Erdös is as follows: if deg _G (u) + deg _G (v) ≥ n + 1 holds for each pair of nonadjacent vertices u and v of G, then G is hamiltonian-connected. In this paper, we improve both theorems by showing that any graph G satisfying the condition in Ore’s theorem is hamiltonian-connected unless G belongs to two exceptional families.

This research was partially supported by the National Science Council of the Republic of China under contract NSC 101-2115-M-033-003-.