Three-coloring triangle-free graphs on surfaces V. Coloring planar graphs with distant anomalies

TL;DR

This paper proves that planar graphs with triangles sufficiently far apart are 3-colorable, extending to a broader class of graphs with specified subgraph constraints and providing conditions for such colorings.

Contribution

It establishes a universal constant for 3-colorability of triangle-free regions in planar graphs with distant triangles, generalizing previous results.

Findings

Existence of a universal constant d for 3-coloring planar graphs with distant triangles.

Generalization to graphs with specified subgraph sets H and distance conditions.

A sufficient condition for 3-coloring with constrained subgraph colorings.

Abstract

We settle a problem of Havel by showing that there exists an absolute constant d such that if G is a planar graph in which every two distinct triangles 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 H be a set of connected subgraphs of G, each of bounded size, such that every two distinct members of H are at least a specified distance apart and all triangles of G are contained in \bigcup{H}. We give a sufficient condition for the existence of a 3-coloring phi of G such that for every B\in H, the restriction of phi to B is constrained in a specified way.