Bicontactual regular maps
This paper is not self-contained, as the reader is directed to a predecessor [the author, Pac. J. Math. 81, 559-568 (1979; Zbl 0433.05021)] six years removed for almost all key concepts. This is particularly unfortunate, as an important part of the author's terminology is not standard. Specifically, a map M is a 2-cell imbedding of a (necessarily connected) graph into a (possibly nonorientable) closed 2-manifold. A flag is a mutual incidence of vertex, edge, and face. If the automorphism group of M is transitive on flags, M is said to be regular (many authors say ''reflexible''). A rotary map has an automorphism group containing all vertex and face rotations (many authors say ''regular'' or ''symmetrical'' here). A rotary map may fail to be regular, in which case it is chiral. In this paper the author classifies all rotary maps with the property that each face meets only one or two others. He shows that all such maps are in fact regular and that they are closed under the action of four operators: D (for dual), P (for Petrie), opp (for opposite), and \(H_ j\) (for \(j^{th}\) order hole) - all defined in the earlier paper. These facts are then used to prove: Every non-trivial rotary map whose number of edges is a power of 2 is orientable.
scientific article; zbMATH DE number 3933098
scientific article