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The study of paths and cycles in graphs has long been fundamental in Graph Theory. One of the most popular questions in tins area is determining when a graph contains a hamiltonian cycle, a cycle containing every vertex of the graph. We call such aMoreThe study of paths and cycles in graphs has long been fundamental in Graph Theory. One of the most popular questions in tins area is determining when a graph contains a hamiltonian cycle, a cycle containing every vertex of the graph. We call such a graph a hamiltonian graph. We will investigate two different aspects of hamiltonian graphs. The first of these is graphs in which the hamiltonian cycle avoids certain subgraphs, and the second is the cycle lengths contained in a hamiltonian graph under given conditions.-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 copy of H in G such that H ≅ F, then we say that G is F-avoiding harniltonian . 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. We also undertake a discussion of F-avoiding pancyclic graphs.-Front there we turn our attention to pancyclic graphs. A graph of order n is said to be pancyclic if it contains cycles of all lengths from three to n. We consider hamiltonian graphs with two vertices of high degree sum. We determine the conditions on this degree sum that assure that the graph is pancyclic. We also consider what cycles must be present, in the graph when the degree suns condition is reduced. Cycle structures in graphs. by Angela K. Harris