The Geng-Hua Fan conditions for pancyclic or Hamilton-connected graphs

Abstract

For a graph G of order n≥3 define P(k), 3≤k≤n, to be the following property: $\text{d}{}_{\mathrm{G}}\text{(x, y) = 2 ⇒}\text{max}\text{{d(x,G), d(y,G)} ≥}\text{k}\text{2}$ for all vertices x and y of G, where d_G(x, y) is the distance between x and y in G. Geng-Hua Fan proved that if G is 2-connected and satisfies P(k), then G contains a cycle of length at least k. In this paper, we prove that if G is 2-connected, $\text{α(G) ≤}\text{n}\text{2}$ and G satisfies P(n−1) [resp. P(n)], then G is Hamiltonian [resp. pancyclic] with some exceptions. We prove also that if G is 3-connected and satisfies P(n+1), then G is Hamilton-connected.