On the genus of order difference interval graph of a finite abelian group

Abstract

The order difference interval graph of a group G, denoted by ΓODI(G), is a graph with V(Γ_ODI(G))=G and two vertices a and b are adjacent in Γ_ODI(G) if and only if o(b) − o(a)∈ [o(a), o(b)]. Without loss of generality, assume that o(a)≤o(b). In this paper, we try to classify all finite abelian groups whose order difference interval graphs are toroidal and projective.