The projective plane crossing numbers of circular graphs*

The authors give an upper bound for the projective plane crossing number of a circular graph. Also, the authors prove the projective plane crossing numbers of circular graph C(8,3) and C(9,3) are 2 and 1, respectively. *This research is supported by the National Natural Science Foundation of China under Grant No. 10671073, Scientific Study Foundation of the Talented People Gathered by Nantong University, partially by Science and Technology Commission of Shanghai Municipality under Grant No. 07XD14011, Shanghai Leading Academic Discipline Project under Grant No. B407, and Natural Science Foundation of Jiangsu’s Universities under Grant No. 07KJB110090.