A quintic nonhamiltonian graph is a quintic graph that is nonhamiltonian. A number of such graphs are illustrated above.
Owens (1980) showed that there exists a nonhamiltonian quintic polyhedral graph on 76 vertices, and van Cleemput and Zamfirescu (2018) demonstrated that there exists such a graph for every even . van Cleemput and Zamfirescu (2018) also demonstrated that the smallest possible such graph must have .
van Cleemput and Zamfirescu (2018) showed that there exists a nonhamiltonian quintic polyhedral graph on vertices for every even , with no better lower bound presently known. Their proof relied on a construction involving the rhombic dodecahedral graph.