Finite groups whose intersection power graphs are toroidal and projective-planar

Huani Li,Xuanlong Ma,Ruiqin Fu

doi:10.1515/math-2021-0071

Huani Li, Xuanlong Ma + Show 1 more

Open Access

https://doi.org/10.1515/math-2021-0071

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Abstract

Abstract The intersection power graph of a finite group G G is the graph whose vertex set is G G , and two distinct vertices x x and y y are adjacent if either one of x x and y y is the identity element of G G , or ⟨ x ⟩ ∩ ⟨ y ⟩ \langle x\rangle \cap \langle y\rangle is non-trivial. In this paper, we completely classify all finite groups whose intersection power graphs are toroidal and projective-planar.

Highlights

A simple graph is an undirected graph without loops and multiple edges
All graphs considered in this paper are finite and simple
An embedding of a graph into a surface is a drawing of the graph on the surface in such a way that its edges may intersect only at their endpoints

Summary

Introduction

A simple graph is an undirected graph without loops and multiple edges. A graph is called finite if its vertex set is finite. The smallest non-negative integer k such that a graph Γ can be embedded on k is called the orientable genus or genus of Γ and is denoted by γ(Γ). The undirected power graph of a group G, denoted (G), is a simple graph whose vertex set is G and two distinct vertices are adjacent if one is a power of the other. In [18], the authors determined the finite groups whose intersection power graphs have book thickness at most two. In [21], the authors classified all finite groups whose power graphs have (non)orientable genus one. In [22], the authors classified all abelian groups whose intersection power graphs have (non)orientable genus one. We completely classify all groups whose intersection power graphs have (non)orientable genus one. I(G) is projective-planar if and only if G ∈ Ψ ∪ { 5, 6, D10, D12, SmallGroup(20, 3)}

Preliminaries

Preliminary results