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). 2009 Wiley Periodicals, Inc. J Graph Theory 64: 1–11, 2010