The Hamiltonian path problem is NP-complete for general graphs. The notion of a follow-up vertex in a permutation graph is introduced in this research, with respect to its Hasse diagram. A necessary and sufficient condition is justified for the existence of a Hamiltonian path in a permutation graph in terms of the existence of a follow-up vertex. Specifically, it is proved that a permutation graph has a Hamiltonian path if and only if it has a follow-up vertex in its lowest layer in the Hasse diagram. A number of improved algorithms are developed using the findings on follow-up vertices to find solutions to the Hamiltonian path problem in permutation graphs that run in \(n(\log (\log n)\) time.