The rainbow $k$--connectivity of the non-commutative graph of a finite group

Abstract

The non-commuting graph Γ( G ) of a non-abelian group G is defined as follows. The vertex set V (Γ( G )) of ℾ( G ) is G Z ( G ) where Z ( G ) denotes the center of G and two vertices x and y are adjacent if and only if xy ≠ yx . We prove that the rainbow k -connectivity of Γ( G ) is equal to ⌈k/2⌉ + 2, for 3 ≤ k ≤ | Z ( G )|.