On the 1-fault hamiltonicity for graphs satisfying Ore's theorem and its generalization

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. The well-known Ore's theorem states that if deg G (u) + deg G (v) ≥ n + k holds for each pair of nonadjacent vertices u and v of G, then G is traceable for k = −1, hamiltonian for k = 0, and hamiltonian-connected for k = 1. Lin et al. generalized Ore's theorem and showed that under the same condition as above, G is r * -connected for 1 ≤ r ≤ k + 2 with k ≥ 1. In this paper, we improve both theorems by showing that the hamiltonicity or r * -connectivity of any graph G satisfying the condition deg G (u) + deg G (v) ≥ n + k with k ≥ −1 is preserved even after one vertex or one edge is removed, unless G belongs to two exceptional families.