Three-coloring triangle-free graphs on surfaces V. Coloring planar graphs with distant anomalies
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.
- Publication:
-
arXiv e-prints
- Pub Date:
- November 2009
- DOI:
- 10.48550/arXiv.0911.0885
- arXiv:
- arXiv:0911.0885
- Bibcode:
- 2009arXiv0911.0885D
- Keywords:
-
- Mathematics - Combinatorics;
- 05C15;
- 05C10
- E-Print:
- 26 pages, no figures. Updated presentation