Necessary condition for a graph to be Non-Hamiltonian
Asking whether a graph is non-Hamiltonian is CoNP-Complete so it is doubtful that such an algorithm exists.

If you are satisfied with a heuristic that provides a certificates of non-Hamiltonicity, see Bondy and Murty's book (page 53) for a sufficient, but not necessary, condition for non-Hamiltonicity:  If $G$ is Hamiltonian then, for every nonempty subset, $S$, of $V$:

$$ \omega(G - S) \le |S| $$

Where $\omega(G - S)$ is the number of connected components left in $G$ after removing the vertices in $S$.  A simple counterexample for the converse is the peterson graph: It passes the above test but is not Hamiltonian.