The Penrose polynomial of a plane graph
type: Item
instance of: scholarly article
title: The Penrose polynomial of a plane graph
zbMATH Open document ID: 0871.05018
author: Martin Aigner
published in: Mathematische Annalen
publication date: 1997-07-06T00:00:00Z
review text: Some 25 years ago Penrose derived several remarkable expressions for the number of 3-edge colorings of a connected 3-regular plane graph \(G\), defining implicitly what is now called the Penrose polynomial \(P(G,\lambda)\). In this paper the Penrose polynomial is studied for arbitrary connected plane graphs. We give an equivalent definition of \(P(G,\lambda)\) in terms of cycles and cocycles and deduce some results on the coefficients and the evaluation of \(P(G,\lambda)\) at integral \(\lambda\). In particular, the Penrose formulae are generalized to arbitrary plane graphs. In addition, connections to the Tutte polynomial, transition systems and the 4-color theorem are discussed.
Mathematics Subject Classification ID: 05C15
Mathematics Subject Classification ID: 05C38
zbMATH DE Number: 1031528
zbMATH Keywords: colorings
zbMATH Keywords: plane graph
zbMATH Keywords: Penrose polynomial
zbMATH Keywords: cycles
zbMATH Keywords: cocycles
zbMATH Keywords: Tutte polynomial
zbMATH Keywords: transition systems
MaRDI profile type: Publication
full work available at URL: s002080050030
OpenAlex ID: W1965359103
DOI: 10.1007/S002080050030
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description: scientific article; zbMATH DE number 1031528