Nonexistence of weakly neighborly polyhedral maps on the orientable 2- manifold of genus 2
A weakly neighborly polyhedral map (w.n.p. map) on a 2-manifold without boundary is a cell-decomposition such that for any two vertices there is a 2-cell containing them. The 2-cells may be n-gons for any \(n\geq 3\). The p-vector \((p_ 3,p_ 4,...)\) consists of the numbers \(p_ n\) of n- gons. The w.n.p. maps on the surfaces of low genus are classified by the authors in several papers. In the present paper they show that there is no w.n.p. map on the orientable surface of genus 2, and they conjecture that this is the only surface with that property. It is well known that every cell-decomposition of the orientable surface of genus 2 must have at least 10 vertices. This is an exceptional behavior because 9 would be the lower bound according to the Heawood formula. From the weak neighborliness the authors get 12 as an upper bound. Then they consider all possible p-vectors for such a hypothetical w.n.p. map and get a contradiction in any case.
scientific article; zbMATH DE number 3938046