A quartic graph is a graph which is 4-regular. The unique quartic graph on five nodes is the complete graph , and the unique quartic graph on six nodes is the octahedral graph. There are two quartic graphs on seven nodes, one of which is the circulant graph . A number of the Archimedean solids have skeletons that are quartic.
The numbers of connected quartic graphs on , 2, ... nodes are 0, 0, 0, 0, 1, 1, 2, 6, 16, 59, ... (OEIS A006820), the numbers of not necessarily connected quartic graphs are 0, 0, 0, 0, 1, 1, 2, 6, 16, 60, ... (OEIS A033301), and the numbers of disconnected quartic graphs for , 11, ... are 1, 1, 3, 8, 25, 88, ... (OEIS A033483; Read and Wilson 1998).
The following table gives a list of some named quartic graphs.
|complete bipartite graph||8||yes|
|rolling cube graph||24||yes|
|small rhombicuboctahedral graph||24||no|
|25-Grünbaum graph 25||25||no|
|small rhombicosidodecahedral graph||60||no|
|(7,3)-bipartite Kneser graph||70||yes|
|5-permutation star graph||120||yes|