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

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)|.