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 ⋃H. We give a sufficient condition for the existence of a 3-coloring ϕ of G such that for every H∈H the restriction of ϕ to H is constrained in a specified way.
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