Sachs triangulations and infinite sequences of regular maps on given type

Abstract

The paper presents a new method which produces a regular map M on an orientable surface via a Sachs triangulation of a permutation group generated by the rotation-involution pair of a certain smaller and simpler irregular map called dessin d'enfant. A regular map M is said to have type { p, q} if p is its vertex degree and q its face size. In Voss (1997), this method has been used in constructing a labelled reflexible regular map of type { p, q} of a closed oriented surface for all face sizes q ⩾ 3 and all vertex degrees p ⩾ 3 q. Here this method is applied to constructing infinite sequences of labelled reflexible regular maps on type { p, q} for all face sizes q ⩾ 3 and all vertex degrees p with p ⩾ q | 4, if q ⩾ 4, and p ⩾ 9, p ≠ 11, if q = 3 (Theorems 4 and 5). Similar methods can be found in Jendrol' et al. (1997) and Archdeacon et al. (1997)