# Tech Reports

## ULCS-10-007

### Colouring Empires in Random Trees (PhD Thesis)

### Abstract

In this thesis we study the empire colouring problem as defined by Percy Heawood in 1890 for maps whose dual planar representation is a tree and have empires formed from exactly r vertices. We define the reduced graph of one such tree as being the graph formed by replacing each empire in the tree by a single vertex such that an edge is present between two vertices u and v of the reduced graph if there was an edge in the original tree joining a vertex belonging to the empire corresponding to u to a vertex belonging to the empire corresponding to v. We then give several results on a number of structural properties of these graphs assuming that the underlying tree is a random labelled tree, and compare them to those of other types of random graphs with the same number of edges. The main contribution of this thesis is a set of results on the worst-case and average-case properties of the empire colourings of trees having an upper bound s on the number of distinct colours used. We first show that if each empire contains exactly r countries, the empire colouring problem can be solved using no more than 2r colours on maps whose dual planar graph is a tree. Furthermore we give an inductive method for building instances that require these many colours. Then, motivated by the results of an empirical investigation of a number of colouring heuristics, we study the proportion of trees on a given number of vertices for which the empire colouring problem can be solved with s leq 2r colours. We prove that for every fixed positive integer r there exists a very precise lower bound on s beneath which almost all trees will admit no r-empire s-colouring. For larger values of s we are then able to give constant positive lower bounds on the probability that s colours are sufficient to colour a random tree.

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