The independence graph of a finite group

Given a finite group G , we denote by Δ ( G ) the graph whose vertices are the elements G and where two vertices x and y are adjacent if there exists a minimal generating set of G containing x and y . We prove that Δ ( G ) is connected and classify the groups G for which Δ ( G ) is a planar graph....

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Bibliographic Details
Published inMonatshefte für Mathematik Vol. 193; no. 4; pp. 845 - 856
Main Author Lucchini, Andrea
Format Journal Article
LanguageEnglish
Published Vienna Springer Vienna 01.12.2020
Springer Nature B.V
Subjects
Apexes
Graph theory
Mathematics
Mathematics and Statistics
20D60
Connectivity
Soluble groups
Planarity
05C25
Generating graph
Generating sets
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Summary:Given a finite group G , we denote by Δ ( G ) the graph whose vertices are the elements G and where two vertices x and y are adjacent if there exists a minimal generating set of G containing x and y . We prove that Δ ( G ) is connected and classify the groups G for which Δ ( G ) is a planar graph.
ISSN:0026-9255
1436-5081
DOI:10.1007/s00605-020-01445-0