The crossing number of hexagonal graph H 3,n in the projective plane
Jing Wang, Junliang Cai, Shengxiang Lv, YuanQiu Huang · Discussiones Mathematicae Graph Theory · 2019
Thomassen described all (except finitely many) regular tilings of the torus S1 and the Klein bottle N2 into (3,6)-tilings, (4,4)-tilings and (6,3)-tilings. Many researchers made great e orts 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 H3,n (n ≥ 2), which is a special kind of (3,6)-tilings, in the projective plane, and prove that crN1(H3,n)={0,n=2,n-1,n≥3.cr{N_1}\left( {{H_{3,n}}} \right) = \left\{ {\matrix{{0,} \hfill & {n = 2,} \hfill \cr {n - 1,} \hfill & {n \ge 3.} \hfill \cr } } \right.