On hamiltonicity of P₃-dominated graphs.

Abstract

We introduce a new class of graphs which we call P₃-dominated graphs. This class properly contains all quasi-claw-free graphs, and hence all claw-free graphs. Let G be a 2-connected P₃-dominated graph. We prove that G is hamiltonian if α(G²) ≤ κ(G), with two exceptions: K₂,₃ and K₁,₁,₃. We also prove that G is hamiltonian, if G is 3-connected and |V(G)| ≤ 5δ(G) − 5. These results extend known results on (quasi-)claw-free graphs.