The projective plane crossing number of a graph is the minimal number of crossings with which the graph can be drawn on the real
projective plane. A graph with projective plane crossing number may be said to
be a projective planar graph.

Richter and Siran (1996) computed the crossing number of the complete bipartite graph on an arbitrary surface. Ho (2005) showed that the projective
plane crossing number of is given by

For ,
2, ..., the first few values are therefore 0, 0, 0, 2, 4, 6, 10, 14, 18, 24, ...
(OEIS A128422).

Richter, R. B. and Širáň, J. "The Crossing Number of in a Surface." J. Graph Th.21, 51-54,
1996.Sloane, N. J. A. Sequence A128422
in "The On-Line Encyclopedia of Integer Sequences."