Vertex-monochromatic connectivity of strong digraphs

A vertex coloring of a strong digraph $D$ is a \emph{strong vertex-monochromatic connection coloring (SVMC-coloring)} if for every pair $u, v$ofverticesin$D$thereexistsan$(u,v)$-path having all its internal vertices of the same color. Let $sm{c}_v(D)$ denote the maximum number of colors used in an SVMC-coloring of a digraph $D$. In this paper we determine the value of $sm{c}_v(D)$, whenever $D$ is the line digraph of a digraph. Also, if $T$ is a tournament, we give conditions to find the exact value of $sm{c}_v(T)$.