Equivelar polyhedra with few vertices (Q5953080)

An (abstract) polyhedral 2-manifold is called equivelar of type \(\{ p,q\}\) if all its 2-faces are \(p\)-gons and all its vertices have degree \(q\). Clearly, all regular polyhedral 2-manifolds are equivelar. In the present paper the authors show that there are precisely 27 simplicial equivelar polyhedral 2-manifolds with \(\leq 11\) vertices. Furthermore, for every \(n\geq -4\) all pairs \((p,q)\) are classified for which there is a \(\{ p,q\}\)-equivelar polyhedral 2-manifold of Euler characteristic \(n\), and explicit constructions are given for five types of equivelar polyhedral 2-manifolds of Euler characteristic \(-2m\), \(m\geq 2\).