Projective Plane Crossing Number
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.
All graphs with graph crossing number 0 or 1 (i.e., planar and singlecross graphs) have projective plane crossing number 0.
Richter and Siran (1996) computed the crossing number of the complete bipartite graph K_(3,n) on an arbitrary surface. Ho (2005) showed that the projective plane crossing number of K_(4,n) is given by
![画像: |_n/3_|[2n-3(1+|_n/3_|)].](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/ProjectivePlaneCrossingNumber/NumberedEquation1.svg){kind=link}
For n=1, 2, ..., the first few values are therefore 0, 0, 0, 2, 4, 6, 10, 14, 18, 24, ... (OEIS A128422).
See also
Projective Planar GraphExplore with Wolfram|Alpha
References
Richter, R. B. and Širáň, J. "The Crossing Number of K_(3,n) in a Surface." J. Graph Th. 21, 51-54, 1996.Sloane, N. J. A. Sequence A128422 in "The On-Line Encyclopedia of Integer Sequences."Referenced on Wolfram|Alpha
Projective Plane Crossing NumberCite this as:
Weisstein, Eric W. "Projective Plane Crossing Number." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ProjectivePlaneCrossingNumber.html