Projective Planar Graph

A graph with projective plane crossing number equal to 0 may be said to be projective planar. Examples of projective planar graphs with graph crossing number >=2 include the complete graph K_6 and Petersen graph P.


Embeddability in the projective plane (i.e., graphs with projective plane crossing number 0) are characterized by a set of exactly 35 forbidden minors (Glover et al. 1979; Archdeacon 1981; Hlinenỳ 2010; Shahmirzadi 2012, p. 7, Fig. 1.1). Note that this set includes the graph unions 2K_(3,3) and 2K_5, each member of which is embeddable in the projective plane. This means that, unlike planar graphs, disjoint unions of graphs which are embeddable in the projective plane may not themselves be embeddable. As of 2022, the plane and projective plane are the only surfaces for which a complete list of forbidden minors is known (Mohar and Škoda 2020).

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


For n=1, 2, ..., the first few values are therefore 0, 0, 0, 2, 4, 6, 10, 14, 18, 24, ... (OEIS A128422).

See also

Forbidden Minor, Projective Plane Crossing Number

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Archdeacon, D. "A Kuratowski Theorem for the Projective Plane." J. Graph Th. 5, 243-246, 1981.Glover, H.; Huneke, J. P.; and Wang, C. S. "103 Graphs That Are Irreducible for the Projective Plane." J. Combin. Th. Ser. B 27, 332-370, 1979.Hlinenỳ, P. "20 Years of Negami's Planar Cover Conjecture." Graphs and Combinatorics 26, 525-536, 2010.Ho, P. T. "The Crossing Number of K_(4,n) on the Real Projective Plane." Disc. Math. 304, 23-33, 2005.Mohar, B. and Škoda, P. "Excluded Minors for the Klein Bottle I. Low Connectivity Case." 1 Feb 2020., R. B. and Širáň, J. "The Crossing Number of K_(3,n) in a Surface." J. Graph Th. 21, 51-54, 1996.Shahmirzadi, A. S. "Minor-Minimal Non-Projective Planar Graphs with an Internal 3-Separation." Ph.D. thesis. Atlanta, GA: Georgia Institute of Technology. Dec. 2012.Sloane, N. J. A. Sequence A128422 in "The On-Line Encyclopedia of Integer Sequences."

Cite this as:

Weisstein, Eric W. "Projective Planar Graph." From MathWorld--A Wolfram Web Resource.

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