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.
The following table summarizes the best (or, in the case of on 
verticvs, best known) smallest quartic
graphs having given crossing number.
 |  | 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 | ? | quartic vertex-transitive graph Qt31 |
| 10 | 14 | ? | regular nonplanar diameter graph (4, 2, 14) |
| 11 | 18? | ? | quartic vertex-transitive graph Qt57 |
| 12 | 16? | ? | quartic vertex-transitive graph Qt44 |
| 13 | 26? | ? | circulant
graph  |
| 14 | 20? | ? | Folkman graph |
| 15 | 20? | ? | circulant graph  ,
533-Haar graph |
