List precoloring extension in planar graphs

TL;DR

This paper investigates conditions under which precolored vertices in planar graphs can be extended to a proper 5-coloring, building on Thomassen's 5-choosability theorem and addressing Albertson's question.

Contribution

The paper extends Thomassen's 5-choosability result to cases with precolored vertices under specific structural conditions in planar graphs.

Findings

Precolored vertices can be extended to a proper 5-coloring if they are sufficiently separated.

The extension is possible when the graph lacks short separating cycles around precolored vertices.

A 'wide' Steiner tree containing all precolored vertices facilitates the extension.

Abstract

A celebrated result of Thomassen states that not only can every planar graph be colored properly with five colors, but no matter how arbitrary palettes of five colors are assigned to vertices, one can choose a color from the corresponding palette for each vertex so that the resulting coloring is proper. This result is referred to as 5-choosability of planar graphs. Albertson asked whether Thomassen's theorem can be extended by precoloring some vertices which are at a large enough distance apart in a graph. Here, among others, we answer the question in the case when the graph does not contain short cycles separating precolored vertices and when there is a "wide" Steiner tree containing all the precolored vertices.