5-list-coloring planar graphs with distant precolored vertices

TL;DR

This paper proves that planar graphs with sufficiently distant precolored vertices can always be 5-list-colored, and provides bounds on critical subgraphs related to 5-list coloring.

Contribution

It establishes the positive answer to Albertson's question and introduces bounds on the size of critical subgraphs for 5-list coloring in planar graphs.

Findings

Planar graphs with distant precolored vertices are 5-list-colorable.

Critical subgraphs have size bounded by the square of the size of the precolored subgraph.

Provides a structural bound on non-colorable subgraphs in planar graphs.

Abstract

We answer positively the question of Albertson asking whether every planar graph can be $5$-list-colored even if it contains precolored vertices, as long as they are sufficiently far apart from each other. In order to prove this claim, we also give bounds on the sizes of graphs critical with respect to 5-list coloring. In particular, if G is a planar graph, H is a connected subgraph of G and L is an assignment of lists of colors to the vertices of G such that |L(v)| >= 5 for every v in V(G)-V(H) and G is not L-colorable, then G contains a subgraph with O(|H|^2) vertices that is not L-colorable.