  The oriented diameter of a bridgeless graph $G$ is $\min\{diam(H)\ | H\ is\
an orientation\ of\ G\}$. A path in an edge-colored graph $G$, where adjacent
edges may have the same color, is called rainbow if no two edges of the path
are colored the same. The rainbow connection number $rc(G)$ of $G$ is the
smallest integer $k$ for which there exists a $k$-edge-coloring of $G$ such
that every two distinct vertices of $G$ are connected by a rainbow path. In
this paper, we obtain upper bounds for the oriented diameter and the rainbow
connection number of a graph in terms of $rad(G)$ and $\eta(G)$, where $rad(G)$
is the radius of $G$ and $\eta(G)$ is the smallest integer number such that
every edge of $G$ is contained in a cycle of length at most $\eta(G)$. We also
obtain constant bounds of the oriented diameter and the rainbow connection
number for a (bipartite) graph $G$ in terms of the minimum degree of $G$.
