Correct nomenclature: Hamilton path, Hamilton's path or Hamiltonian path?
I have seen both Hamilton path (or Hamilton cycle) and Hamiltonian path (or Hamiltonian cycle). A graph, however, is always Hamiltonian (if it contains a Hamilton/Hamiltonian cycle). Consider for example the titles of two papers: Hamiltonian cycles in random regular graphs (Fenner &amp; Frieze) and Generating and counting Hamilton cycles in random regular graphs (Frieze, Jerrum, Molloy, Robinson &amp; Wormald). Alan Frieze apparently doesn't care too strongly about the issue.

However, I've never seen Hamilton's path or cycle.