The rainbow connection number of the power graph of a finite group

TL;DR

This paper investigates the rainbow connection number of power graphs of finite groups, providing exact values for certain classes and bounds for others, enhancing understanding of graph connectivity in algebraic structures.

Contribution

It determines the rainbow connection number for power graphs of finite groups with maximal involutions or nilpotency, and establishes an upper bound for groups without maximal involutions.

Findings

Rainbow connection number is at most three for groups without maximal involutions.

Exact rainbow connection numbers are found for groups with maximal involutions or nilpotent groups.

Some nonnilpotent groups' power graphs' rainbow connection numbers are also characterized.

Abstract

This paper studies the rainbow connection number of the power graph $\Gamma_G$ of a finite group $G$. We determine the rainbow connection number of $\Gamma_G$ if $G$ has maximal involutions or is nilpotent, and show that the rainbow connection number of $\Gamma_G$ is at most three if $G$ has no maximal involutions. The rainbow connection numbers of power graphs of some nonnilpotent groups are also given.