At Crossing Number Graphs are minimal cubic graphs with crossing numbers 0 to 8. At Smallest Cubic Crossing Number Graph are some corrections -- 8D and 8E can be drawn with less than 8 crossings, and a conjectured 10 crossing graph only needed 9 crossings.
Crossing Number Web Compute and QuickCross can calculate crossing numbers.
As of 8 April 2019, we know of
2 26V cubic graphs needing 9 crossings.
2 28V cubic graphs needing 10 crossings.
1 28V cubic graphs needing 11 crossings.
0 30V cubic graphs needing 12 crossings.
1 30V cubic graphs needing 13 crossings.
Are there other cubic graphs that can be added to the list above? Based on the counts of lower graphs, it is suspected that these graphs will be rare.
Sequence A307450.
Of the 545 girth 7 30V graphs, only one might have CN 12, according to Michael Haythorpe. I don't have a 12-crossing embedding.
