The smallest quartic graphs with graph crossing number 
have been termed "crossing number graphs"
or 
-crossing
graphs by Pegg and Exoo (2009) in the case of smallest
cubic crossing number graphs.
Some smallest quartic crossing number graphs are also transmission-minimal regular graphs.
The numbers of 
-vertex connected quartic graphs having 0, 1, ... crossings for 
,
2, ... are given by
 |
(OEIS A390644).
The following table summarizes the smallest quartic graphs having given crossing number. For 
, 1, 2, ..., there are 1, 1, 1, 5, 1, 1, 14, 32, 1, 123,
9, ?, 1, ... (OEIS A389263) distinct crossing
number graphs, the first few of which are illustrated above. The number of nodes
in the smallest quartic graph with crossing numbers
,
1, ... are 6, 5, 7, 9, 8, 10, 12, 13, 12, 14, 14, 16, 16, ?, 16, ... (OEIS A389265).
Here, the computations giving values up to 16 crossings were completed by E. Weisstein
on Feb. 10, 2026.
 |  | count |  |
| 0 | 6 | 1 | octahedral graph  |
| 1 | 5 | 1 | pentatope graph  |
| 2 | 7 | 1 | co- |
| 3 | 9 | 5 | circulant
graph  , generalized
quadrangle  , and 3 others |
| 4 | 8 | 1 | complete
bipartite graph  |
| 5 | 10 | 1 | circulant graph  |
| 6 | 12 | 14 | Chvátál graph, circulant
graph  , quartic
vertex-transitive graph Qt23, and 11 others |
| 7 | 13 | 32 | 13-cyclotomic graph and 31 others |
| 8 | 12 | 1 | circulant graph  |
| 9 | 14 | 3 | quartic vertex-transitive graph Qt31 and two others |
| 10 | 14 | 1 | 1 graph |
| 11 | 16 | 123 | 123 graphs |
| 12 | 16 | 9 | quartic vertex-transitive graph Qt44 and at least one other |
| 13 | 17 |  | at least 7 graphs |
| 14 | 16 | 1 | 1 graph |
| 15 | 17? | ? |
