You Canʹt Paint Yourself into a Corner

Thomassen posed the following problem: SupposeGis a planar graph andW⊆V(G) such that the distance between any two vertices inWis at least 100. Can a 5-coloring ofWbe extended to a 5-coloring ofG? It is known that no 4-coloring result of this nature can hold. We provide a best possible solution to Thomassenʹs problem as well as some generalizations.