The rainbow $k$-connectivity of two classes of graphs

A path in an edge-colored graph $G$, where adjacent edges may be colored the same, is called a rainbow path if no two edges of $G$ are colored the same. For a $\kappa$-connected graph $G$ and an integer $k$ with $1\leq k\leq \kappa$, the rainbow $k$-connectivity $rc_k(G)$ of $G$ is defined as the minimum integer $j$ for which there exists a $j$-edge-coloring of $G$ such that every two distinct vertices of $G$ are connected by $k$ internally disjoint rainbow paths. Let $G$ be a complete $(\ell+1)$-partite graph with $\ell$ parts of size $r$ and one part of size $p$ where $0\leq p <r$ (in the case $p=0$, $G$ is a complete $\ell$-partite graph with each part of size $r$). This paper is to investigate the rainbow $k$-connectivity of $G$. We show that for every pair of integers $k\geq 2$ and $r\geq 1$, there is an integer $f(k,r)$ such that if $\ell\geq f(k,r)$, then $rc_k(G)=2$. As a consequence, we improve the upper bound of $f(k)$ from $(k+1)^2$ to $ck^{{3/2}}+C$, where $0<c<1$, $C=o(k^{{3/2}})$, and $f(k)$ is the integer such that if $n \geq f(k)$ then $rc_k(K_n)=2$.