Polyhedral graphs with restricted number of faces of the same type
type: Item
instance of: scholarly article
title: Polyhedral graphs with restricted number of faces of the same type
zbMATH Open document ID: 0995.05039
DOI: 10.1016/S0012-365X(01)00103-0
published in: Discrete Mathematics
publication date: 2002-05-21T00:00:00Z
review text: A polyhedral graph is a planar and three-connected graph \(G=(V,E,F)\) with vertex set \(V\), edge set \(E\) and face set \(F\). A face \(\alpha \in F\) is an \(\langle a_1,\dots,a_l\rangle\)-face if \(\alpha\) is \(l\)-gon and the degrees of the vertices \(x_1,\dots,x_l\) incident with \(\alpha\) in cyclic order are \(a_1,\dots,a_l\), respectively. The type of a face \(\alpha \in F\) is the lexicographic minimum \(\langle b_1,\dots,b_l\rangle\) such that \(\alpha\) is an \(\langle b_1,\dots,b_l\rangle\)-face. A polyhedral graph \(G\) is called oblique if no two its faces are of the same type. Let \(z_{\langle b_1,\dots,b_l\rangle}\) be the number of faces of type \(\langle b_1,\dots,b_l\rangle\) in \(G\) and \(z\) be a positive integer. A polyhedral graph \(G\) is called \(z\)-oblique if \(z_{\langle b_1,\dots,b_l\rangle}\leq z\) for all types of faces of \(G\). The main result of the paper states that for each \(z\), the set of \(z\)-oblique graphs is finite. A consequence of this result is that for any positive integer there exists a polyhedral graph having at least \(k\) faces of the same type. The proof of the result is based on the discharging method.
Mathematics Subject Classification ID: 05C10
Mathematics Subject Classification ID: 05C75
zbMATH DE Number: 1743045
zbMATH Keywords: polyhedral graphs
zbMATH Keywords: oblique graphs
author: Margit Voigt
author: Hansjoachim Walther
reviewed by: Gabriel Semanišin
MaRDI profile type: Publication
full work available at URL: s0012-365x(01)00103-0
OpenAlex ID: W2080823368
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description: scientific article; zbMATH DE number 1743045