Rainbow connectivity of the non-commuting graph of a finite group

TL;DR

This paper investigates the rainbow connectivity properties of the non-commuting graph of finite non-abelian groups, establishing that its rainbow connection number is 2 and demonstrating the existence of infinitely many groups with this property for any k.

Contribution

It proves that the rainbow connection number of the non-commuting graph of any finite non-abelian group is 2, and shows infinitely many such groups for any k.

Findings

Rainbow 2-connection number of the graph is 2

Rainbow connection number of the graph is 2

Existence of infinitely many groups with rainbow k-connection number 2 for any k

Abstract

Let $G$ be a finite non-abelian group. The non-commuting graph $\Gamma_G$ of $G$ has the vertex set $G\setminus Z(G)$ and two distinct vertices $x$ and $y$ are adjacent if $xy\ne yx$, where $Z(G)$ is the center of $G$. We prove that the rainbow $2$-connectivity of $\Gamma_G$ is $2$. In particular, the rainbow connection number of $\Gamma_G$ is $2$. Moreover, for any positive integer $k$, we prove that there exist infinitely many non-abelian groups $G$ such that the rainbow $k$-connectivity of $\Gamma_G$ is $2$.