# Three-coloring triangle-free graphs on surfaces V. Coloring planar graphs with distant anomalies - Mathematics > Combinatorics

Abstract: We settle a problem of Havel by showing that there exists an absoluteconstant d such that if G is a planar graph in which every two distincttriangles are at distance at least d, then G is 3-colorable.In fact, we prove a more general theorem. Let G be a planar graph, and let Hbe a set of connected subgraphs of G, each of bounded size, such that every twodistinct members of H are at least a specified distance apart and all trianglesof G are contained in \bigcup{H}. We give a sufficient condition for theexistence of a 3-coloring phi of G such that for every B\in H, the restrictionof phi to B is constrained in a specified way.

Author: Zdenek Dvorak, Daniel Kral, Robin Thomas

Source: https://arxiv.org/