Orientador: Prof. Dr. Letícia Rodrigues Bueno / Dissertação (mestrado) - Universidade Federal do ABC, Programa de Pós-Graduação em Ciência da Computação, 2015. / Given a graph G, the hamiltonian cycle problem consists in determining if there is a cycle containing all vertices of G exactly once. This problem is known to be NP-Complete, therefore a recent trend is to searching for long cycles in order to determine the cycle with the largest possible number of vertices. Another trend is searching for related structures. In this aspect, being prism-hamiltonian has been an interesting relaxation of being hamiltonian. The prism over a graph G consists of two copies of G with an edge joining the corresponding vertices. A graph G is prism-hamiltonian if the prism over G contains a hamiltonian cycle. In this work, we study a conjecture which claims that every 4-connected 4-regular graph is prism-hamiltonian. We prove the conjecture for claw-free graphs. In fact, for a subclass of claw-free 4-connected 4-regular graphs, we prove a stronger result: its hamiltonicity; therefore, corroborating to another conjecture from 1993 which states that claw-free 4-connected 4-regular graphs are hamiltonian. Given a graph G, let G1 = GK2 and Gq = Gq..1K2, for q > 1. For every connected graph G, we prove that Gq is hamiltonian for q dlog2 (G)e, where (G) is the maximum degree of G.