Precoloring extension involving pairs of vertices of small distance

TL;DR

This paper explores how the presence of pairs of vertices at small distances within a precolored subset affects the extension of graph colorings, generalizing previous results that required larger distances between such pairs.

Contribution

It extends Albertson's coloring extension theorem to cases where pairs of vertices are at distance at most three or two, analyzing the impact on the number of colors needed.

Findings

Pairs of vertices at distance at most three influence coloring extension.

Pairs of vertices at distance at most two further restrict coloring options.

Generalizes previous results by Albertson and Moore on coloring extensions.

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 $P\subset V(G)$. 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].