A quartic nonhamiltonian graph is a quartic graph that is nonhamiltonian. A number of such graphs are illustrated above.

Van Cleemput and Zamfirescu (2018) gave a 39-vertex example of a nonhamiltonian quartic polyhedral graph (the 39-van Cleemput-Zamfirescu graph), showed that there exists a on vertices for each , and conjectured that gives the smallest such graph possible. Van Cleemput and Zamfirescu (2018) also gave a 78-node quartic nonhamiltonian polyhedral graph that is untraceable.