  A path in an edge-colored graph $G$ is called monochromatic if any two edges
on the path have the same color. For $k\geq 2$, an edge-colored graph $G$ is
said to be monochromatic $k$-edge-connected if every two distinct vertices of
$G$ are connected by at least $k$ edge-disjoint monochromatic paths, and $G$ is
said to be uniformly monochromatic $k$-edge-connected if every two distinct
vertices are connected by at least $k$ edge-disjoint monochromatic paths such
that all edges of these $k$ paths colored with a same color. We use $mc_k(G)$
and $umc_k(G)$ to denote the maximum number of colors that ensures $G$ to be
monochromatic $k$-edge-connected and, respectively, $G$ to be uniformly
monochromatic $k$-edge-connected. In this paper, we first conjecture that for
any $k$-edge-connected graph $G$,
$mc_k(G)=e(G)-e(H)+\lfloor\frac{k}{2}\rfloor$, where $H$ is a minimum
$k$-edge-connected spanning subgraph of $G$. We verify the conjecture for
$k=2$. We also prove the conjecture for $G=K_{k+1}$ when $k\geq4$ is even, and
for $G=K_{k,n}$ when $k\geq4$ is even, or when $k=3$ and $n\geq k$. When $G$ is
a minimal $k$-edge-connected graph, we give an upper bound of $mc_k(G)$, i.e.,
$mc_k(G)\leq k-1$, and $mc_k(G)\leq \lfloor\frac{k}{2}\rfloor$ when
$G=K_{k,n}$. For the uniformly monochromatic $k$-edge-connectivity, we prove
that for all $k$, $umc_k(G)=e(G)-e(H)+1$, where $H$ is a minimum
$k$-edge-connected spanning subgraph of $G$.
