  A path in a vertex-colored graph is called a \emph{vertex-monochromatic path}
if its internal vertices have the same color. A vertex-coloring of a graph is a
\emph{monochromatic vertex-connection coloring} (\emph{MVC-coloring} for
short), if there is a vertex-monochromatic path joining any two vertices in the
graph. For a connected graph $G$, the \emph{monochromatic vertex-connection
number}, denoted by $mvc(G)$, is defined to be the maximum number of colors
used in an \emph{MVC-coloring} of $G$. These concepts of vertex-version are
natural generalizations of the colorful monochromatic connectivity of
edge-version, introduced by Caro and Yuster. In this paper, we mainly
investigate the Erd\H{o}s-Gallai-type problems for the monochromatic
vertex-connection number $mvc(G)$ and completely determine the exact value.
Moreover, the Nordhaus-Gaddum-type inequality for $mvc(G)$ is also given.
