On the chromatic number of the plane for map-type colorings
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915135925256192 |
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| author | Sokolov, Georgy Voronov, Vsevolod |
| author_facet | Sokolov, Georgy Voronov, Vsevolod |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_01958 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the chromatic number of the plane for map-type colorings Sokolov, Georgy Voronov, Vsevolod Combinatorics 05C15 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. |
| title | On the chromatic number of the plane for map-type colorings |
| topic | Combinatorics 05C15 |
| url | https://arxiv.org/abs/2502.01958 |