Hamilton cycles in claw-heavy graphs

A graph G on n ≥ 3 vertices is called claw-heavy if every induced claw ( K 1 , 3 ) of G has a pair of nonadjacent vertices such that their degree sum is at least n . In this paper we show that a claw-heavy graph G has a Hamilton cycle if we impose certain additional conditions on G involving numbers of common neighbors of some specific pair of nonadjacent vertices, or forbidden induced subgraphs. Our results extend two previous theorems of Broersma, Ryjáček and Schiermeyer [H.J. Broersma, Z. Ryjáček, I. Schiermeyer, Dirac’s minimum degree condition restricted to claws, Discrete Math. 167–168 (1997) 155–166], on the existence of Hamilton cycles in 2-heavy graphs.