Nonplanar graphs and well-covered cycles
type: Item
instance of: scholarly article
title: Nonplanar graphs and well-covered cycles
zbMATH Open document ID: 0732.05033
DOI: 10.1016/0012-365X(90)90024-C
author: R. B. Richter
published in: Discrete Mathematics
publication date: 1990-01-01T00:00:00Z
review text: If G is planar then the collection \({\mathcal C}\) of face boundaries of a planar embedding satisfy the following properties: (i) each edge of G is in exactly two members of \({\mathcal C}\); and (ii) for each spanning tree T of G, at least two members of \({\mathcal C}\) have all but one edge in T. Rosenfeld conjectured in 1988 that a 2-connected graph with a set of cycles \({\mathcal C}\) satisfying (i) and (ii) is planar. This note furnishes a counterexample to Rosenfeld's conjecture, which is isomorphic to the line graph of the Petersen graph. That it satisfies (ii) is easily shown by looking at its geometric dual.
Mathematics Subject Classification ID: 05C38
Mathematics Subject Classification ID: 05C10
zbMATH DE Number: 4210173
zbMATH Keywords: planar graph
zbMATH Keywords: covering cycles
reviewed by: Norman F. Quimpo
MaRDI profile type: Publication
cites work: On the cycle space of graphs
cites work: The cycle space of an embedded graph
cites work: Embeddings of graphs with no short noncontractible cycles
cites work: Q4273853
Wikidata QID: Q127629415
full work available at URL: 0012-365x(90)90024-c
OpenAlex ID: W1978883959
description: scientific article; zbMATH DE number 4210173