  Let $G$ be a connected undirected graph.~A vertex coloring $f$ of $G$ is an
$N_i$-vertex coloring if for each vertex $x$ in $G$, the number of different
colors assigned to $N_G(x)$ is at most $i$.~The $N_i$-chromatic number of $G$,
denoted by $t_i(G)$, is the maximum number of colors which are used in an
$N_i$-vertex coloring of $G$.
  In this paper, we provide sharp bounds for $t_i(G)$ of a graph $G$ in terms
of its vertex cover number, maximum degree and diameter, respectively. We also
determine precise values for $t_i(G)$ in some cases.
