A note on regular maps with large planar width

The planar width of a finite, non-spherical map is the smallest number of intersections of a non-contractible closed curve on the supporting surface of the map with the embedded graph. We study the norm of the minimal polynomial of 2 cos(pi/n) to improve the existing upper bounds on the size of the smallest regular map of type {m, n} whose planar width is larger than r.