Hamiltonian graphs involving neighborhood unions

Dirac proved that a graph G is hamiltonian If the minimum degree delta(G) >= n/2, where n is the order of G. Let G be a graph and A subset of V(G). The neighborhood of A is N(A) = {b: ab is an element of E(G) for some a is an element of A}. For any positive integer k, we show that every (2k - 1)-connected graph of order n >= 16 k(3) is hamiltonian if \N(A)\ >= n/2 for every independent vertex Set A of k vertices. The result contains a few known results as special cases. The case of k = 1 is the classic result of Dirac when n is large and the case of k = 2 is a result of Broersma, Van den Heuvel, and Veldman when n is large. For general k, this result improves a result of Chen and Liu. The lower bound 2k - 1 on connectivity is best possible in general while the lower bound 16k(3) for n is conjectured to be unnecessary. (C) 2006 Wiley Periodicals, Inc.