Solutions to conjectures on the $(k,\ell)$-rainbow index of complete graphs

The $(k,\ell)$-rainbow index $rx_{k, \ell}(G)$ of a graph $G$ was introduced by Chartrand et. al. For the complete graph $K_n$ of order $n\geq 6$, they showed that $rx_{3,\ell}(K_n)=3$ for $\ell=1,2$. Furthermore, they conjectured that for every positive integer $\ell$, there exists a positive integer $N$ such that $rx_{3,\ell}(K_{n})=3$ for every integer $n \geq N$. More generally, they conjectured that for every pair of positive integers $k$ and $\ell$ with $k\geq 3$, there exists a positive integer $N$ such that $rx_{k,\ell}(K_{n})=k$ for every integer $n \geq N$. This paper is to give solutions to these conjectures.