Power Graph Classes and Overfullness

TL;DR

This paper characterizes when the power graph of a finite group is overfull or Class 2, revealing that this occurs precisely for cyclic groups of odd prime power order, thus classifying these power graphs.

Contribution

It provides a complete characterization of overfull and Class 2 power graphs of finite groups, linking these properties to the group's cyclic structure of odd prime power order.

Findings

Power graphs are overfull if and only if the group is cyclic of odd prime power order.

Power graphs are Class 2 if and only if the group is cyclic of odd prime power order.

The classification of power graphs is achieved through group-theoretic properties.

Abstract

In this paper, we investigate the edge-coloring number of the power graph of a finite group. We characterize which finite groups have overfull power graphs, showing that this occurs if and only if the group is cyclic of odd prime power order. We then show that this classsifies the class of power graphs: The power graph of a finite group is Class 2 if and only if the group is cyclic of odd prime power order.