Forbidden Subgraphs of Prime Order Element Graph

TL;DR

This paper investigates the structural properties of prime-order element graphs of finite groups, characterizing when these graphs are perfect, cograph, chordal, claw-free, or interval graphs.

Contribution

It provides forbidden subgraph characterizations for prime-order element graphs of finite groups, a novel analysis linking group theory and graph theory.

Findings

Identifies conditions for $ ext{Gamma}(G)$ to be perfect

Determines when $ ext{Gamma}(G)$ is a cograph or chordal

Characterizes $ ext{Gamma}(G)$ as claw-free or interval graph

Abstract

In this paper, we study different forbidden subgraph characterizations of the prime-order element graph $\Gamma(G)$ defined on a finite group $G$. Its set of vertices is the group $G$ and two vertices $x,y \in G$ are adjacent if the order of $xy$ is prime. More specifically, we investigate the conditions when $\Gamma(G)$ is perfect, cograph, chordal, claw-free, and interval graph.