In this paper, we show that every 3-connected claw-free graph such that every induced cycle of length at least 4 has at most 8 edges contained in a triangle is hamiltonian. This implies that every 3-connected {K1,3} ∪ {Ci |i ≥ 9}-free graph is hamiltonian. We also show that every 3-connected o-heavy graph whose induced cycle of length is at most 8 is… (More)