An edge-colored graph $G$, where adjacent edges may have the same color, is
{\it rainbow connected} if every two vertices of $G$ are connected by a path
whose edge has distinct colors. A graph $G$ is {\it $k$-rainbow connected} if
one can use $k$ colors to make $G$ rainbow connected. For integers $n$ and $d$
let $t(n,d)$ denote the minimum size (number of edges) in $k$-rainbow connected
graphs of order $n$. Schiermeyer got some exact values and upper bounds for
$t(n,d)$. However, he did not get a lower bound of $t(n,d)$ for $3\leq
d<\lceil\frac{n}{2}\rceil$. In this paper, we improve his lower bound of
$t(n,2)$, and get a lower bound of $t(n,d)$ for $3\leq
d<\lceil\frac{n}{2}\rceil$.Comment: 8 page