The Complexity of the Empire Colouring Problem

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 $r \geq 3$.Comment: 23 pages, 12 figure

Complexity Results for the Empire Problem in Collection of Stars

International audienceIn this paper, we study the Empire Problem, a generalization of the coloring problem to maps on two-dimensional compact surface whose genus is positive. Given a planar graph with a certain partition of the vertices into blocks of size r, for a given integer r, the problem consists of deciding if s colors are sufficient to color the vertices of the graph such that vertices of the same block have the same color and vertices of two adjacent blocks have different colors. In this paper, we prove that given a 5-regular graph, deciding if there exists a 4-coloration is NP-complete. Also, we propose conditional NP-completeness results for the Empire Problem when the graph is a collection of stars. A star is a graph isomorphic to K 1,q for some q ≥ 1. More exactly, we prove that for r ≥ 2, if the (2r − 1)-coloring problem in 2r-regular connected graphs is NP-complete, then the Empire Problem for blocks of size r + 1 and s = 2r − 1 is NP-complete for forests of K 1, r . Moreover, we prove that this result holds for r = 2. Also for r ≥ 3, if the r-coloring problem in (r + 1)-regular graphs is NP-complete, then the Empire Problem for blocks of size r + 1 and s = r is NP-complete for forests of K 1, 1 = K 2, i.e., forest of edges. Additionally, we prove that this result is valid for r = 2 and r = 3. Finally, we prove that these results are the best possible, that is for smallest value of s or r, the Empire Problem in these classes of graphs becomes polynomial

Martingales on trees and the empire chromatic number of random trees

Abstract. We study the empire colouring problem (as defined by Percy Heawood in 1890) for maps whose dual planar graph is a tree, with empires formed by exactly r countries. We prove that, for each fixed r&gt; 1, with probability approaching one as the size of the graph grows to infinity, the minimum number of colours for which a feasible solution exists takes one of seven possible values.