  For integers $k, r > 0$, 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 at least $\min\{r, d(v)\}$ differently colored
vertices. Given $r$, the smallest integer $k$ for which $G$ has a conditional
$(k,r)$-coloring is called the $r$th order conditional chromatic number
$\chi_r(G)$ of $G$. We give results (exact values or bounds for $\chi_r(G)$,
depending on $r$) related to the conditional coloring of some graphs. We
introduce \emph{unique conditional colorability} and give some related results.
(Keywords. cartesian product of graphs; conditional chromatic number; gear
graph; join of graphs.)
