Hamiltonian cycles avoiding sets of edges in a graph

A spanning cycle in a graph G is called a hamiltonian cycle, and if such a cycle exists G is said to be hamiltonian. Let G be a graph and H be a subgraph of G. If G contains a hamiltonian cycle C such that E(C) \ E(H) is empty, we say that C is an H-avoiding hamiltonian cycle. Let F be any graph. If G contains an H-avoiding hamiltonian cycle for every subgraph H of G such that H = F, then we say that G is F-avoiding hamiltonian. In this paper, we give minimum degree and degree-sum conditions which assure that a graph G is F-avoiding hamiltonian for various choices of F. In particular, we consider the cases where F is a union of k edge-disjoint hamiltonian cycles or a union of k edge-disjoint perfect matchings. If G is F-avoiding hamiltonian for any such F, then it is possible to extend families of these types in G. Finally, we undertake a discussion of F-avoiding pancyclic graphs.