Eulerian properties of non-commuting and non-cyclic graphs of finite groups. Issue 6 (3rd June 2018)

• Record Type: Journal Article
• Authors: Costa, Daniel; Davis, Veronica; Gill, Kenneth; Hinkle, Gerhardt; Reid, Les
• Is Part Of: Communications in algebra. Volume 46:Issue 6(2018)
• Page Start: 2659
• Page End: 2665
• Publication Date: 2018-06-03
• Subjects: Eulerian -- group -- non-commuting graph -- non-cyclic graph -- path-eulerian; Primary: 20D60 -- Secondary: 05C25; Algebra -- Periodicals; 512.005
• DOI: 10.1080/00927872.2017.1392534
• Languages: English
• ISSNs: 0092-7872

Abstract

ABSTRACT: For a non-abelian group G, the non-commuting graph Γ ( G ) has G − Z ( G ) as its vertex set and two vertices x and y are connected by an edge if xy ≠ yx . For a non-cyclic group G, the non-cyclic graph Γ ′ ( G ) has G −Cyc( G ) as its vertex set, where Cyc( G ) = { x |⟨ x, y ⟩ is cyclic, for all y ∈ G } and two vertices x and y are connected by an edge if ⟨ x, y ⟩ is not cyclic. We show that for all finite non-abelian groups G, Γ ( G ) is eulerian if and only if Γ ′ ( G ) is eulerian. We investigate the eulerian properties of these graphs for various G showing, in particular, that Γ ( G ) (and hence Γ ′ ( G ) ) is path-eulerian if and only if G ≅ S 3 .