Hamilton cycles in regular 3-connected graphs

This result is best possible for k = 3 since the Petersen graph is a nonhamiltonian, 2-connected, 3-regular graph on 10 vertices. It is essentially best possible for k > 4 since there exist non-hamiltonian, 2-connected, kregular graphs on 3k + 4 vertices for k even, and 3k + 5 vertices for all k. Examples of such graphs are given in [ 1, 3 1. The problem of determining the values of k for which all 2-connected, k-regular graphs on n vertices are hamiltonian was first suggested by G. Szekeres. Erdijs and Hobbs [ 3 ] proved that such graphs are hamiltonian if n < 2k + ck”*, where c is a positive constant. Subsequently, Bollobas and Hobbs [ 1 ] showed that G is hamiltonian if n < +k. We shall in fact prove a result slightly stronger than Theorem 1.