Hamiltonian cycles in n-factor-critical graphs

A graph G is said to be n-factor-critical if G − S has a 1-factor for any S ⊂V (G) with |S| = n. In this paper, we prove that if G is a 2-connected n-factor-critical graph of order p with 3(G)¿ 2 (p−n−1), then G is hamiltonian with some exceptions. To extend this theorem, we de6ne a (k; n)-factor-critical graph to be a graph G such that G − S has a k-factor for any S ⊂V (G) with |S| = n. We conjecture that if G is a 2-connected (k; n)-factor-critical graph of order p with 3(G)¿ 2 (p− n− k), then G is hamiltonian with some exceptions. In this paper, we characterize all such graphs that satisfy the assumption, but are not 1-tough. Using this, we verify the conjecture for k62. c © 2001 Elsevier Science B.V. All rights reserved.