The Complexity of the Empire Colouring Problem

TL;DR

This paper studies the computational complexity of the empire colouring problem, providing polynomial-time solutions under certain conditions and proving NP-hardness in various cases for maps with specific graph structures.

Contribution

It offers a complete complexity characterization for empire colouring on trees and partial results for planar graphs, advancing understanding of the problem's computational boundaries.

Findings

Polynomial-time solvability when no induced subgraph of high average degree exists.

NP-hardness for empire colouring on trees when s is between 3 and 2r-1.

NP-hardness for planar graphs with fewer than 7 colours for r=2, and fewer than 6r-3 for r ≥ 3.

Abstract

We investigate the computational complexity of the empire colouring problem (as defined by Percy Heawood in 1890) for maps containing empires formed by exactly $r > 1$ countries each. We prove that the problem can be solved in polynomial time using $s$ colours on maps whose underlying adjacency graph has no induced subgraph of average degree larger than $s/r$. However, if $s \geq 3$, the problem is NP-hard even if the graph is a forest of paths of arbitrary lengths (for any $r \geq 2$, provided $s < 2r - \sqrt(2r + 1/4+ 3/2)$. Furthermore we obtain a complete characterization of the problem's complexity for the case when the input graph is a tree, whereas our result for arbitrary planar graphs fall just short of a similar dichotomy. Specifically, we prove that the empire colouring problem is NP-hard for trees, for any $r \geq 2$, if $3 \leq s \leq 2r-1$ (and polynomial time solvable otherwise). For arbitrary planar graphs we prove NP-hardness if $s<7$ for $r=2$, and $s < 6r-3$, for $r \geq 3$. The result for planar graphs also proves the NP-hardness of colouring with less than 7 colours graphs of thickness two and less than $6r-3$ colours graphs of thickness $r \geq 3$.