  A path in an edge-colored graph is called a monochromatic path if all edges
of the path have a same color. We call $k$ paths $P_1,\cdots,P_k$ rainbow
monochromatic paths if every $P_i$ is monochromatic and for any two $i\neq j$,
$P_i$ and $P_j$ have different colors. An edge-coloring of a graph $G$ is said
to be a rainbow monochromatic $k$-edge-connection coloring (or $RMC_k$-coloring
for short) if every two distinct vertices of $G$ are connected by at least $k$
rainbow monochromatic paths. We use $rmc_k(G)$ to denote the maximum number of
colors that ensures $G$ has an $RMC_k$-coloring, and this number is called the
rainbow monochromatic $k$-edge-connection number. We prove the existence of
$RMC_k$-colorings of graphs, and then give some bounds of $rmc_k(G)$ and
present some graphs whose $rmc_k(G)$ reaches the lower bound. We also obtain
the threshold function for $rmc_k(G(n,p))\geq f(n)$, where
$\lfloor\frac{n}{2}\rfloor> k\geq 1$.
