The Crossing Number of Hexagonal Graph H3,n in the Projective Plane

Abstract

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.

Keywords

• projective plane
• crossing number
• hexagonal graph
• drawing
• 05c10
• 05c62