Minimal quadrangulations of orientable surfaces
A polyhedron on a closed orientable 2-manifold S is called a quadrangulation if each face of the polyhedron is bounded by a 4-cycle (in the graph which is the 1-skeleton of the polyhedron) and the intersection of any two such 4-cycles is either empty or at most one edge and at most three vertices. A quadrangulation of S is said to be minimal for S if no quadrangulation of S has fewer faces. The number of faces in a minimal quadrangulation of S is denoted by \(\psi\) (S). For example, it is shown that the cube gives the unique minimal quadrangulation of the sphere \(S_ 0\); thus \(\psi (S_ 0)=6\). Current graphs and rotation schemes are used to construct quadrangulations by the complete graphs \(K_{8t+5}\) (t\(\geq 0)\) and by the general octahedral graphs \(K_{n(2)}\) (the complement of a 1-factor in \(K_{2n}\); here, \(n\geq 3)\). These are all minimal (except for \(K_{3(2)}\) on \(S_ 1)\), so that \(\psi (S_{8t^ 2+5t+1)}=16t^ 2+18t+5\) (t\(\geq 0)\) and \(\psi (S_{(n- 1)/2)})=n(n-1)\) (n\(\geq 4)\).
scientific article; zbMATH DE number 4087674