Nonexistence of weakly neighborly polyhedral maps on the orientable 2- manifold of genus 2
type: Item
instance of: scholarly article
title: Nonexistence of weakly neighborly polyhedral maps on the orientable 2- manifold of genus 2
zbMATH Open document ID: 0585.57004
DOI: 10.1016/0097-3165(86)90008-7
author: Amos Altshuler
author: Ulrich Brehm
published in: Journal of Combinatorial Theory. Series A
publication date: 1986-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 57N05
Mathematics Subject Classification ID: 57Q15
zbMATH DE Number: 3938046
zbMATH Keywords: weakly neighborly polyhedral map
zbMATH Keywords: 2-manifold
zbMATH Keywords: cell-decomposition
reviewed by: Wolfgang Kühnel
MaRDI profile type: Publication
full work available at URL: 0097-3165(86)90008-7
OpenAlex ID: W1995905306
cites work: The weakly neighborly polyhedral maps on the non-orientable 2-manifold with Euler characteristic -2
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cites work: Minimal triangulations on orientable surfaces
cites work: Q4773723
description: scientific article; zbMATH DE number 3938046