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 σ3(G)⩾32(p−n−1), then G is hamiltonian with some exceptions. To extend this theorem, we define 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 σ3(G)⩾32(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 k⩽2.