Forbidden triples implying Hamiltonicity: for all graphs

In [2], Brousek characterizes all triples of graphs, G₁, G₂, G₃, with ${\displaystyle G_i=K_{1,3}}$ for some i = 1, 2, or 3, such that all G₁G₂G₃-free graphs contain a hamiltonian cycle. In [6], 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 G₁, G₂, G₃-free graphs of sufficiently large order contain a hamiltonian cycle. In this paper, a characterization will be 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.