The Analysis Of K₇ In The Projective Plane N₁

Crossing number theory is a branch of graph theory and has many applications. For in-stance, we can apply it to the design of electronic circuits, the graphical representation of DNA in biological engineering and so on. Many results have been found and mainly concentrate on the crossing number in the plane, but few people research on the crossing number in the non-planar.In this paper, we mainly concentrate on the crossing number of K7 in the projective plane Ni. Starting from sub-graph K5 and K6 and applying the Jordan curve theorem for plane graphs, together with topological methods of the projective plane, we show that the crossing number of K7 in the projective plane N1 is 3.