Bilinski maps and construction of geodesics
by
Mark E. Watkins
Syracuse University, Syracuse NY, USA
Coauthors: Jennifer Bruce (Maryville College, Maryville TN, USA)

Let G_{a, b} denote the class of 1-ended, connected planar maps wherein the valence of each vertex is finite but at least a while the covalence of each face is finite but at least b. We consider also the subclasses G_{a+, b} (resp., G_{a, b+}) wherein no edge is incident with two a-valent vertices (resp., two b-covalent faces).

The classes G_{4, 4}, G_{6, 3}, and G_{3, 6} span this chasm, while maps in G_{5, 3+} and its dual class have exponential growth. We study the members of these classes from the following two points of view:

1. Are they representable as Bilinski maps all of whose edges either lie on concentric circuits about an arbitrary vertex or join vertices on consecutive circuits?

2. Is there a simple algorithm, starting from an arbitrary edge, for constructing a geodetic path of arbitrary length, a geodetic ray, or even a geodetic double ray?

Date received: October 24, 2000