  For integers $k>0$ and $0<r \leq \Delta$ (where $r \leq k$), a conditional
$(k,r)$-coloring of a graph $G$ is a proper $k$-coloring of the vertices of $G$
such that every vertex $v$ of degree $d(v)$ in $G$ is adjacent to vertices with
at least $\min\{r, d(v)\}$ differently colored neighbors. The smallest integer
$k$ for which a graph $G$ has a conditional $(k,r)$-coloring is called the
$r$th order conditional chromatic number, denoted by $\chi_r(G)$. For different
values of $r$ we first give results (exact values or bounds for $\chi_r(G)$
depending on $r$) related to the conditional coloring of graphs. Then we obtain
$\chi_r(G)$ of certain parameterized graphs viz., windmill graph, line graph of
windmill graph, middle graph of friendship graph, middle graph of a cycle, line
graph of friendship graph, middle graph of complete $k$-partite graph, middle
graph of a bipartite graph and gear graph. Finally we introduce \emph{unique
conditional colorability} and give some related results.
