Every 3-connected Claw-free Z₈-Free graph is hamiltonian

In this article, we first show that every 3-edge-connected graph with circumference at most 8 is supereulerian, which is then applied to show that a 3-connected claw-free graph without Z₈ as an induced subgraph is Hamiltonian, where Z₈ denotes the graph derived from identifying one end vertex of P₉ (a path with 9 vertices) with one vertex of a triangle. The above two results are both best possible in a sense that the number 8 cannot be replaced by 9 and they also extend former results by Brousek et al. in (Discrete Math 196 (1999), 29-50) and by Luczak and Pfender in (J Graph Theory 47 (2004), 111-121).