Geodesic
The Crossing Number of Hexagonal Graph H3,n in the Projective Plane
Discussiones Mathematicae. Graph Theory, Tome 42 (2022) no. 1, pp. 197-218

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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 ≥ 2), which is a special kind of (3,6)-tilings, in the projective plane, and prove that cr_N_1 (H_3,n) = 0, amp; n=2, n-1, amp; n ≥ 3.
Keywords: projective plane, crossing number, hexagonal graph, drawing 
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Wang, Jing; Cai, Junliang; Lv, Shengxiang; Huang, Yuanqiu. The Crossing Number of Hexagonal Graph H3,n in the Projective Plane. Discussiones Mathematicae. Graph Theory, Tome 42 (2022) no. 1, pp. 197-218. http://geodesic.mathdoc.fr/item/DMGT_2022_42_1_a12/