Empires Make Cartography Hard: The Complexity of the Empire Colouring Problem

We study 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 >= 3, the problem is NP-hard for forests of paths of arbitrary lengths (if s < r) for trees (if r >= 2 and s < 2r) and arbitrary planar graphs (if s < 7 for r = 2, and s < 6r - 3, for r >= 3). The result for trees shows a perfect: dichotomy (the problem is NP-hard if 3 <= s <= 2r-1 and polynomial time solvable otherwise). The one for planar graphs 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 >= 3.