# How many ways to color the map of America?

## TL;DR

This paper explores the number of valid colorings of geographic maps of Canada, France, and the USA with four or fewer colors, providing new insights and results related to graph chromatic polynomials.

## Contribution

It introduces the first computations of map colorings for real-world countries and presents new findings on the chromatic polynomial of graphs.

## Key findings

- Computed the number of four-colorings for maps of Canada, France, and the USA.
- Discovered new results on the chromatic polynomial of graphs.
- Extended understanding of map coloring complexities.

## Abstract

Although the Four Color Conjecture originated in cartography, surprisingly, there is nothing in the literature on the number of ways to color an actual geographic map with four or fewer colors. In this paper, we compute these numbers, with exponentially increasing order of difficulty, for the maps of Canada, France, and the USA. Our attempts to compute the latter two lead to some new results on the chromatic polynomial of graphs.