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 $\mathcal{H}$ be a set of connected subgraphs of G, each of bounded size, such that every two distinct members of $\mathcal{H}$ are at least a specified distance apart and all triangles of G are contained in $\bigcup \mathcal{H}$. We give a sufficient condition for the existence of a 3-coloring ϕ of G such that for every $H\in \mathcal{H}$ the restriction of ϕ to H is constrained in a specified way.

中文翻译：

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曲面V上的三色无三角图。对具有远距异常的平面图进行着色

通过显示存在一个绝对常数d来解决Havel的问题，如果G是一个平面图，其中每两个不同的三角形之间的距离至少为d，则G是3色的。实际上，我们证明了一个更一般的定理。令G为平面图，令$\mathcal{H}$是G的一组连通子图的集合，每个子图的大小是有界的，使得G的每两个不同成员$\mathcal{H}$至少相距指定距离，并且G的所有三角形都包含在$\bigcup \mathcal{H}$。我们给出一个充分条件3着色的存在φ的ģ使得对于每$H\in \mathcal{H}$的限制φ到ħ以指定的方式被约束。