Path partition number in tough graphs (Q1356718)

Let \(\pi(G)\) be the smallest integer \(t\) such that there exists a system of pairwise disjoint paths \(P_1,\dots,P_t\) containing all vertices of \(G\) if \(G\) is not Hamiltonian, \(\pi(G)\) be \(-1\) if \(G\) Hamiltonian-connected, and \(\pi(G)\) be 0 if \(G\) is only Hamiltonian. \(\pi(G)\) is called the path partition number of \(G\). Let \(\tau(G)\) be the toughness of \(G\); see, i.e., \textit{V. Chvátal}, Discrete Math. 5, 215-228 (1973; Zbl 0256.05122). The author proves that there exists equivalence between: (a) Every 2-tough graph on at least 3 vertices is Hamiltonian. (b) There exists an integer \(c\) such that \(c\geq\pi(G)\) if \(\tau(G)>2\).