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 numbers of 
-vertex connected quartic graphs having 0, 1, ... crossings for 
, 2, ... are given by
 |
(OEIS A390644).
The following table summarizes the best (or, in the case of on 
vertices, best known) smallest quartic graphs having given crossing number.
For 
= 0, 1, 2, ..., there are 1, 1, 1, 5, 1, 1, 14, 32, 1, ... (OEIS A389263) distinct crossing number graphs, illustrated above. The number of nodes in the smallest quartic graph with crossing number 
, 1, ... are 6, 5, 7, 9, 8, 10, 12, 13, 12, 14, 14, ... (OEIS A389265).
 |  | 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? | ? | |
| 12 | 16 | ? | quartic vertex-transitive graph Qt44 |
| 13 | 17? | ? | |
| 14 | 17? | ? | |
| 15 | 17? | ? |
