We prove that a k-connected graph (k>2) is Hamiltonian if it is not contractible to one of a specified collection of graphs of order 2k+1. The theorem generalizes a previous result of the authors. The proof partly parallels that of the following, less general, result of Chvátal and Erdös: A k-connected graph containing no independent set of more than k points (k>2) is Hamiltonian (*). Also stated in terms of contractibility are sufficient conditions for graphs to be traceable, Hamiltonian connected or 1-Hamiltonian, respectively. Conditions analogous to (*) guaranteeing the same properties were found by Chvátal and Erdös and by Lesniak. For traceable and 1-Hamiltonian graphs the contraction theorems sharpen the corresponding analogues of (*), while equivalence is conjectured for Hamiltonian connected graphs.