Enhanced power graphs of groups are weakly perfect

TL;DR

This paper proves that the enhanced power graph of any finite group is weakly perfect, with its clique and chromatic numbers equal to the maximum order of an element, using combinatorial methods.

Contribution

It establishes that enhanced power graphs of finite groups are weakly perfect and characterizes their clique and chromatic numbers.

Findings

Enhanced power graphs are weakly perfect.

Clique and chromatic numbers equal maximum element order.

Provides combinatorial proof and related graph remarks.

Abstract

A graph is weakly perfect if its clique number and chromatic number are equal. We show that the enhanced power graph of a finite group $G$ is weakly perfect: its clique number and chromatic number are equal to the maximum order of an element of $G$. The proof requires a combinatorial lemma. We give some remarks about related graphs.