HAMILTONIAN GRAPHS INVOLVING DISTANCES

Let G be a graph of order n. We show that if G is a 2-connected graph and max{d(u),d(upsilon)} + \N(u) or N(upsilon)\ greater-than-or-equal-to n for each pair of vertices u, upsilon at distance two, then either G is hamiltonian or G congruent-to 3K(n/3) or T1 or T2, where n = 0 (mod 3), and T1 and T2 are the edge sets of two vertex disjoint triangles containing exactly one vertex from each K(n/3). This result generalizes both Fan's and Lindquester's results as well as several others.