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

TL;DR

This paper advances the understanding of the chromatic number of the plane for map-like colorings, proving that at least 7 colors are necessary under certain boundary conditions, extending previous lower bounds.

Contribution

It establishes a new lower bound of 7 colors for coloring maps with complex boundaries, improving upon the prior bound of 6 colors.

Findings

At least 7 colors are required for certain map colorings.

The result applies to arbitrary polygons with specific boundary conditions.

The proof adapts techniques from distance-interval chromatic number problems.

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.