  In this paper, we consider coloring of graphs under the assumption that some
vertices are already colored. Let $G$ be an $r$-colorable graph and let
$P\subset V(G)$. Albertson [J.\ Combin.\ Theory Ser. B \textbf{73} (1998),
189--194] has proved that if every pair of vertices in $P$ have distance at
least four, then every $(r+1)$-coloring of $G[P]$ can be extended to an
$(r+1)$-coloring of $G$, where $G[P]$ is the subgraph of $G$ induced by $P$. In
this paper, we allow $P$ to have pairs of vertices of distance at most three,
and investigate how the number of such pairs affects the number of colors we
need to extend the coloring of $G[P]$. We also study the effect of pairs of
vertices of distance at most two, and extend the result by Albertson and Moore
[J.\ Combin.\ Theory Ser. B \textbf{77} (1999) 83--95].
