On weakly neighborly polyhedral maps of arbitrary genus
A weakly neighborly polyhedral map (w.n.p. map) is a 2-dimensional cell complex which decomposes a 2-manifold without boundary, such that for every two vertices there is a 2-cell containing them. It is shown that the boundary complexes of the triangular prism and of any pyramid over a polygon are the only w.n.p maps on the surface of genus O. For every other genus g (orientable or nonorientable) the number of w.n.p. maps is shown to be finite, and the number V(g) of vertices is estimated where the upper bound grows as \((4g)^{2/3}\) for \(g\to \infty\). This is based on various equations and inequalities between the numbers of facets \(f_ i\) having \(k_ i\) vertices. The authors gave a classification of w.n.p. maps on surfaces of low genus: for the torus [Geom. Dedicata 18, 227-238 (1985; Zbl 0567.51021)] which admits five w.n.p. maps, for the orientable surface of genus 2 [J. Comb. Theory, Ser. A 42, 87-103 (1986; Zbl 0585.57004)] which does not admit such a w.n.p. map, and for all nonorientable surfaces with Euler characteristic \(\chi\geq -2\) including the Möbius band (to appear in several papers).
scientific article; zbMATH DE number 3951435