A Hamilton cycle in a graph is a cycle that passes through every vertex of the graph. A graph is called Hamiltonian if it contains such a cycle. In this thesis we investigate two classes of graphs, defined by local criteria. Graphs in these classes, with a simple set of exceptions K, were proven to be Hamiltonian by Asratian, Broersma, van den Heuvel, and Veldman in 1996 and by Asratian in 2006, respectively. We prove here that in addition to being Hamiltonian, graphs in these classes have stronger cyclic properties. In particular, we prove that if a graph G belongs to one of these classes, then for each vertex x in G there is a sequence of cycles such that each cycle contains the vertex x, and the shortest cycle in the sequence has length at most 5; the longest cycle in the sequence is a Hamilton cycle (unless G belongs to the set of exceptions K, in which case the longest cycle in the sequence contains all but one vertex of G); each cycle in the sequence except the first contains all vertices of the previous cycle, and at most two other vertices. Furthermore, for each edge e in G that does not lie on a triangle, there is a sequence of cycles with the same three properties, such that each cycle in the sequence contains the edge e.