Eulerian properties of non-commuting and non-cyclic graphs of finite groups. Issue 6 (3rd June 2018) - Costa, Daniel; Davis, Veronica; Gill, Kenneth; Hinkle, Gerhardt; Reid, Les
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 .