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, equivelar polyhedra. 2000 Mathematics Subject Classification. Primary: 51M20; Secondary: 52B15 1 Introduction Combinatorially regular polyhedra in Euclidean 3-space E3 are polyhedral embeddings in E3 of regular maps (cell-complexes) on orientable compact closed surfaces. Regular maps on surfaces have been studied from combinatorial, topological, algebraic and geometric viewpoints for well over 100 years (see Coxeter & Moser [9]). Combinatorially regular polyhedra and their underlying topological maps can generally be viewed as higher-genus analogues of the Platonic

. M. and Wills, J. M.). Elsevier Publishers, 1993, pp. 535–554 [5] Bagchi B., Datta B.: A structure theorem for pseudomanifolds. Discr. Math. 168 (1998), 41–60 [6] Broughton S. A.: Classifying finite group actions on surfaces of low genus. J. Pure Appl. Algebra 69 (1990), 233–270 [7] Croom F. H.: Basic Concepts of Algebraic Topology. Springer-Verlag, New York— Heidelberg 1978 [8] Datta B., Nilakantan N.: Equivelar polyhedra with few vertices. Discrete & Comput. Geom. 26 (2001), 429–461 [9] Datta B., Upadhyay A. K.: Degree-regular triangulations of torus and Klein