On Tutte polynomials and cycles of plane graphs
type: Item
instance of: scholarly article
title: On Tutte polynomials and cycles of plane graphs
zbMATH Open document ID: 0595.05033
DOI: 10.1016/0095-8956(88)90083-4
author: François Jaeger
published in: Journal of Combinatorial Theory. Series B
publication date: 1988-01-01T00:00:00Z
review text: Let t(G,x,y) be the Tutte polynomial (or dichromate) of the connected plane graph G. It is known (Martin; Las Vergnas) that t(G,x,x) can be expressed in terms of the family of partitions of the edge-set of the medial graph M(G) of G into non-crossing cycles. Moreover, t(G,-1,-1) can be expressed in terms of the number of crossing cycles of M(G) (Martin; Rosenstiehl and Read). Another result of Penrose on the number of edge-3-colorings of a cubic connected plane graph G can be viewed as an evaluation of t(G,0,-3) in terms of the family of partitions of the edge-set of M(G) into cycles avoiding certain transitions. We unify and generalize these results by giving expressions of t(G,x,y) in terms of cycle partitions of M(G) for all x, y such that (x-1)(y-1)\(\neq 0\) or \(x=y=1\).
Mathematics Subject Classification ID: 05C15
Mathematics Subject Classification ID: 05C38
zbMATH DE Number: 3957141
zbMATH Keywords: Tutte polynomial
zbMATH Keywords: plane graph
zbMATH Keywords: number of crossing cycles
zbMATH Keywords: number of 3- edge colourings
zbMATH Keywords: cubic connected plane graph
MaRDI profile type: Publication
full work available at URL: 0095-8956(88)90083-4
OpenAlex ID: W2016645081
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description: scientific article; zbMATH DE number 3957141