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.
