Potential forbidden triples implying hamiltonicity: for sufficiently large graphs

In [1], Brousek characterizes all triples of connected graphs, G₁,G₂,G₃, with ${\displaystyle G_i=K_{1,3}}$ for some i = 1,2, or 3, such that all G₁G₂ G₃-free graphs contain a hamiltonian cycle. In [8], Faudree, Gould, Jacobson and Lesniak consider the problem of finding triples of graphs G₁,G₂,G₃, none of which is a ${\displaystyle K_{1,s}}$, s ≥ 3 such that G₁G₂G₃-free graphs of sufficiently large order contain a hamiltonian cycle. In [6], a characterization was given of all triples G₁,G₂,G₃ with none being ${\displaystyle K_{1,3}}$, such that all G₁G₂G₃-free graphs are hamiltonian. This result, together with the triples given by Brousek, completely characterize the forbidden triples G₁,G₂,G₃ such that all G₁G₂G₃-free graphs are hamiltonian. In this paper we consider the question of which triples (including ${\displaystyle K_{1,s}}$, s ≥ 3) of forbidden subgraphs potentially imply all sufficiently large graphs are hamiltonian. For s ≥ 4 we characterize these families.