Problems on pairs of trees and the four colour problem of planar graphs

Abstract

We described a new characterization of the Four-Colour Problem in terms of its equivalence to a problem of 3-edge colouring pairs of binary trees each with m leaves with the restriction that for every i, 1<-i≤m, edges adjacent to the i-th leaf have the same colour in both trees. This problem is equivalent to non-trivial subclasses of many problems in mathematics and computer science of which we described three. These provide new and enticing opportunities in the search for shorter proofs of the Four-Colour Theorem and efficient algorithms for Four-Colouring. Conversely, taking the polynomial time solution for Four-Colouring, our equivalences provide unexpected polynomial time solutions for non-trivial sub-classes of problems for which in general only exponential time algorithms are known. The reductions between the various problems were shown to be rapid (in at worst O(nα(n, n)) time) and are of interest in themselves. It is likely, because of the nature of the problem of Colouring Pairs of Trees, that many other non-trivial subclasses of important problems defined on trees will find unexpected polynomial time solutions.