Hamiltonian cycles avoiding sets of edges in a graph

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 ensure 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. IfG 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.