Toughness and hamiltonicity in almost claw-free graphs

Some known results on claw-free (K 1;3-free) graphs are generalized to the larger class of almost claw-free graphs which were introduced by Ryjj a cek. In particular, we show that a 2-connected almost claw-free graph is 1-tough, and that a 2-connected almost claw-free graph on n vertices is hamiltonian if 1 3 (n ? 2), thereby (partly) generalizing results of Matthews and Sumner. Finally, we use a result of Bauer et al. to show that a 2-connected almost claw-free graph on n vertices is hamiltonian if d(u) + d(v) + d(w) n for all independent sets of vertices u, v and w.