A sufficient local degree condition for Hamiltonicity in locally finite claw-free graphs

Among the well-known sufficient degree conditions for the Hamiltonicity of a finite graph, the condition of Asratian and Khachatrian is the weakest and thus gives the strongest result. Diestel conjectured that it should extend to locally finite infinite graphs  GG, in that the same condition implies that the Freudenthal compactification of GG contains a circle through all its vertices and ends. We prove Diestel’s conjecture for claw-free graphs.