In the terminology of Robertson et al. (1997), a doublecross graph is a graph having a drawing in the plane
with two crossings incident with a common graph face.
Edwards et al. (2016) used the same terminology. This drawing condition is
not equivalent to having graph crossing number
2.
The Robertson-Edwards terminology is directly related to the Tutte three-edge-coloring conjecture. Robertson et al. (1997) showed that the
conjecture reduces to the apex and doublecross cases,
and Edwards et al. (2016) proved the doublecross case.
The numbers of doublecross simple graphs on nodes are 0, 0, 0, 0, 0, 1, 39, ...,
and the numbers of connected graphs are 0, 0,
0, 0, 0, 1, 38, ....
Edwards, K.; Sanders, D. P.; Seymour, P.; and Thomas, R. "Three-Edge-Colouring Doublecross Cubic Graphs." J. Combin. Th. Ser.
B119, 66-95, 2016. https://doi.org/10.1016/j.jctb.2015.12.006.Robertson,
N.; Seymour, P. D.; and Thomas, R. "Tutte's Edge-Colouring Conjecture."
J. Combin. Th. Ser. B70, 166-183, 1997. https://doi.org/10.1006/jctb.1997.1752.Robertson,
N.; Seymour, P. D.; and Thomas, R. "Girth Six Cubic Graphs Have Petersen
Minors." Combinatorica39, 1413-1423, 2019.
nodes are 0, 0, 0, 0, 0, 1, 39, ...,
and the numbers of connected graphs are 0, 0,
0, 0, 0, 1, 38, ....
