Empire Maps on Surfaces

TL;DR

This paper explores the mathematical theory of map coloring, focusing on empire maps on various surfaces, and proves Heawood's Empire Conjecture in a new case, advancing understanding of coloring problems on complex surfaces.

Contribution

It introduces new results on empire maps on higher genus surfaces and proves Heawood's Empire Conjecture in an previously unverified case.

Findings

Proves Heawood's Empire Conjecture for a new class of surfaces

Summarizes known results for empire maps on higher genus surfaces

Discusses recent contributions to map coloring theory

Abstract

This report is an introduction to mathematical map colouring and the problems posed by Heawood in his paper of 1890. There will be a brief discussion of the Map Colour Theorem; then we will move towards investigating empire maps in the plane and the recent contributions by Wessel. Finally we will conclude with a discussion of all known results for empire maps on higher genus surfaces and prove Heawoods Empire Conjecture in a previously unknown case.