Hamilton cycles in claw-heavy graphs

A graph GG on n≥3n≥3 vertices is called claw-heavy if every induced claw (K1,3K1,3) of GG has a pair of nonadjacent vertices such that their degree sum is at least nn. In this paper we show that a claw-heavy graph GG has a Hamilton cycle if we impose certain additional conditions on GG 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.