Edge-oblique polyhedral graphs.
type: Item
instance of: scholarly article
title: Edge-oblique polyhedral graphs.
zbMATH Open document ID: 1037.05017
DOI: 10.1016/S0166-218X(03)00447-5
author: Jens Schreyer
author: Hansjoachim Walther
published in: Discrete Applied Mathematics
publication date: 2004-03-14T00:00:00Z
review text: Let \(G=(V,E,F)\) be a polyhedral graph (i.e. 3-connected plane graph) with vertex set \(V\), edge set \(E\) and face set \(F\). An edge \(e\) is defined to be of type \((a,b; m,n)\) if it is incident with an \(a\)-vertex and a \(b\)-vertex and with an \(m\)-face and an \(n\)-face. Here an \(x\)-vertex and an \(r\)-face denote a vertex of degree \(x\) and a face of size \(r\), respectively. A graph which contains no two edges of the same type is called edge-oblique and if it contains at most \(z\) edges of each type it is called \(z\)-edge-oblique. In this paper it is proved that the set of edge-oblique graphs is finite. Several properties of edge-oblique graphs are proved. For example, it is shown that if \(G\) is an oblique graph then its maximum degree \(\Delta(G)\) is at most \(30\) and it has less than \(880\) edges. Another nice result of the paper states that the set of \(z\)-edge-oblique graphs is finite too. Note that an analogous problem concerning face-oblique graphs is considered in \textit{M. Voigt} and \textit{H. Walther} [Discrete Math. 244, 473--478 (2002; Zbl 0995.05039)].
Mathematics Subject Classification ID: 05C10
Mathematics Subject Classification ID: 52B10
Mathematics Subject Classification ID: 05C35
zbMATH DE Number: 2055511
zbMATH Keywords: plane graph
zbMATH Keywords: type of edge
zbMATH Keywords: oblique graphs
reviewed by: Stanislav Jendroľ
MaRDI profile type: Publication
cites work: Some news about oblique graphs
cites work: On the simplicial 3-polytopes with only two types of edges
cites work: Convex 3-polytopes with exactly two types of edges
cites work: Polyhedral graphs with restricted number of faces of the same type
cites work: Polyhedral graphs with extreme numbers of types of faces
full work available at URL: s0166-218x(03)00447-5
OpenAlex ID: W2086332155
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description: scientific article; zbMATH DE number 2055511