A smallest quartic crossing number graph is a finite, simple, connected quartic graph of positive order whose vertex count is minimum among such graphs with the same graph crossing number. The name is used here by analogy with the smallest cubic crossing number graphs of Pegg and Exoo (2009).
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 graph crossing number.
For 
,
1, 2, ..., there are 1, 1, 1, 5, 1, 1, 14, 32, 1, 3, 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 for graphs on up to 16 vertices 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 8 others |
| 13 | 17 |  | at least 7 graphs |
| 14 | 16 | 1 | 1 graph |
| 15 | 17? | ? |
