A result on combinatorial curvature for embedded graphs on a surface
type: Item
instance of: scholarly article
title: A result on combinatorial curvature for embedded graphs on a surface
zbMATH Open document ID: 1161.05031
published in: Discrete Mathematics
publication date: 2009-01-28T00:00:00Z
review text: Let \(G\) be an infinite graph embedded in a surface such that each open face of the embedding is homeomorphic to an open disk and is bounded by finite number of edges. For each vertex \(x\) of \(G\) the combinatorial curvature is defined as \[ K_G(x)=1-\frac{d(x)}2+\sum\limits_{\sigma\in F(x)}\frac1{|\sigma|}\,, \] where \(d(x)\) is the degree of \(x,F(x)\) is the multiset of all open faces \(\sigma\) in the embedding such that the closure \(\bar{\sigma}\) contains \(x\), and \(|\sigma|\) is the number of sides of edges bounding the face \(\sigma\). In this paper, for a finite simple graph \(G\) embedded in a surface with \(3\leq d_G(x)<\infty\) and \(K_G(x)>0\) for all \(x\in V(G)\), it is proved that {\parindent=7mm \begin{itemize}\item[(i)]if \(G\) is embedded in a projective plane and \(|V(G)|=n\geq 290\), then \(G\) is isomorphic to \(P_n\); \item[(ii)]if \(G\) is embedded in a sphere and \(|V(G)|=n\geq 580\), then \(G\) is isomorphic to either the graph of an \(n\)-sided prism a \(A_n\) or an \(n\)-sided antiprisma \(B_n\). \end{itemize}}
Mathematics Subject Classification ID: 05C10
zbMATH DE Number: 5499799
zbMATH Keywords: Combinatorial curvature
zbMATH Keywords: Gauss-Bonnet formula
zbMATH Keywords: Euler relation
zbMATH Keywords: Infinite graph
zbMATH Keywords: Embedding
zbMATH Keywords: Face cycle
zbMATH Keywords: Finiteness theorem
author: Lili Zhang
reviewed by: Stanislav Jendroľ
MaRDI profile type: Publication
full work available at URL: j.disc.2007.11.007
OpenAlex ID: W2082676938
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DOI: 10.1016/J.DISC.2007.11.007
description: scientific article; zbMATH DE number 5499799