  An edge-coloured graph $G$ is rainbow connected if there exists a rainbow
path between any two vertices. A graph $G$ is said to be $k$-rainbow connected
if there exists an edge-colouring of $G$ with at most $k$ colours that is
rainbow connected. For integers $n$ and $k$, let $t(n,k)$ denote the minimum
number of edges in $k$-rainbow connected graphs of order $n$. In this note, we
prove that $t(n,k) = \lceil k(n-2)/(k-1) \rceil$ for all $n, k \ge 3$
