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arXiv · 2502.01958

On the chromatic number of the plane for map-type colorings

Georgy Sokolov·Vsevolod Voronov
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Abstract

We consider the Hadwiger-Nelson problem on the chromatic number of the plane under conditions of coloring a map containing a finite number of vertices in any bounded region. Woodall (1973) and Townsend (1981) showed that at least 6 colors would be required. In the present paper, it is shown that at least 7 colors are required to color a map in which the boundaries are not arcs of a unit circle and three boundaries connect at each vertex. As a corollary, we obtain that at least 7 colors are required for a proper coloring in which the regions are arbitrary polygons. The proof relies on techniques developed for a similar result concerning the chromatic number of the plane with a forbidden interval of distances.

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Georgy Sokolov, Vsevolod Voronov. 2025-02-04. On the chromatic number of the plane for map-type colorings. https://arxiv.org/abs/2502.01958

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