Chvátal-Erdös type theorems

The Chvátal-Erdös theorems imply that if G is a graph of order n ≥ 3 with κ(G) ≥ α(G), then G is hamiltonian, and if κ(G) > α(G), then G is hamiltonian-connected. We generalize these results by replacing the connectivity and independence number conditions with a weaker minimum degree and independence number condition in the presence of sufficient connectivity. More specifically, it is noted that if G is a graph of order n and k ≥ 2 is a positive integer such that κ(G) ≥ k, δ(G) > (n + k − k)/(k + 1), and δ(G) ≥ α(G) + k − 2, then G is hamiltonian. It is shown that if G is a graph of order n and k ≥ 3 is a 246 J.R. Faudree, R.J. Faudree, R.J. Gould, ... positive integer such that κ(G) ≥ 4k +1, δ(G) > (n+k −2k)/k, and δ(G) ≥ α(G) + k − 2, then G is hamiltonian-connected. This result supports the conjecture that if G is a graph of order n and k ≥ 3 is a positive integer such that κ(G) ≥ k, δ(G) > (n + k − 2k)/k, and δ(G) ≥ α(G) + k − 2, then G is hamiltonian-connected, and the conjecture is verified for k = 3 and 4.