  A path in an edge-colored graph is called {\em rainbow} if no two edges of it
are colored the same. For an $\ell$-connected graph $G$ and an integer $k$ with
$1\leq k\leq \ell$, the {\em rainbow $k$-connection number} $rc_k(G)$ of $G$ is
defined to be the minimum number of colors required to color the edges of $G$
such that every two distinct vertices of $G$ are connected by at least $k$
internally disjoint rainbow paths. Fujita et. al. proposed a problem that what
is the minimum constant $\alpha>0$ such that for all 2-connected graphs $G$ on
$n$ vertices, we have $rc_2(G)\leq \alpha n$. In this paper, we prove that
$\alpha=1$ and $rc_2(G)=n$ if and only if $G$ is a cycle of order $n$, settling
down this problem.
