  A path in an edge-colored graph is called a \emph{monochromatic path} if all
the edges on the path are colored the same. An edge-coloring of $G$ is a
\emph{monochromatic connection coloring} (MC-coloring, for short) if there is a
monochromatic path joining any two vertices in $G$. The \emph{monochromatic
connection number}, denoted by $mc(G)$, is defined to be the maximum number of
colors used in an MC-coloring of a graph $G$. These concepts were introduced by
Caro and Yuster, and they got some nice results. In this paper, we will study
two kinds of Erd\H{o}s-Gallai-type problems for $mc(G)$, and completely solve
them.
