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? | ? |
