The Crossing Number of Hexagonal Graph H_3,n in the Projective Plane

Thomassen described all (except finitely many) regular tilings of the torus $ S_1 $ and the Klein bottle $N_2$ into (3,6)-tilings, (4,4)-tilings and (6,3)-tilings. Many researchers made great efforts to investigate the crossing number of the Cartesian product of an $m$-cycle and an $n$-cycle, which is a special kind of (4,4)-tilings, either in the plane or in the projective plane. In this paper we study the crossing number of the hexagonal graph $ H_{3,n} (n \ge 2) $, which is a special kind of (3,6)-tilings, in the projective plane, and prove that $$ cr_{N_1} (H_{3,n}) = \begin{cases} 0, & n=2, \\ n-1, & n \ge 3. \end{cases} $$