Precoloring extension involving pairs of vertices of small distance

Abstract

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 $P\subset V\left(G\right)$. Albertson (1998) has proved that if every pair of vertices in $P$ has distance at least four, then every $(r+1)$-coloring of $G\left[P\right]$ can be extended to an $(r+1)$-coloring of $G$, where $G\left[P\right]$ 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 $G\left[P\right]$. We also study the effect of pairs of vertices of distance at most two, and extend the result by Albertson and Moore (1999).