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We first show that if a graph G of order n contains a hamiltonian path connecting two nonadjacent vertices u and v such that d(u) + d(v) ≥ n, then G is pancyclic. By using this result, we prove that if G is hamiltonian with order n ≥ 20 and if G has two nonadjacent vertices u and v such that d(u) + d(v) ≥ n + z, where z = 0 when n is odd and z = 1… (More)

A set S is an offensive alliance if for every vertex v in its boundary N (S) − S it holds that the majority of vertices in v's closed neighbourhood are in S. The offensive alliance number is the minimum cardi-nality of an offensive alliance. In this paper we explore the bounds on the offensive alliance and the strong offensive alliance numbers (where a… (More)

In the class of k-connected claw-free graphs, we study the stability of some hamilto-nian properties under a closure operation introduced by the third author. We prove that (i) the properties of pancyclicity, vertex pancyclicity and cycle extendability are not stable for any k (i.e., for any of these properties there is an innnite family of graphs G k of… (More)