Edge-Colouring Pairs of Binary Trees: Towards a Concise Proof of the Four-Colour Theorem of Planar Maps

Abstract

The famous Four-colour Problem (FCP) of planar maps is equivalent, by an optimally fast reduction, to the problem of Colouring Pairs of Binary Trees (CPBT). Extant proofs of FCP lack conciseness, lucidity and require hours of electronic computation. The search for a satisfactory proof continues and, in this spirit, we explore two approaches to CPBT. In the first, we prove that a satisfactory proof exists if the rotational path between the two trees of the problem instance always satisfies a specific condition embodied in our Shortest Path Conjecture. In our second approach, we look for patterns of colourability within regular forms of tree pairs and seek to understand all instances of CPBT as a perturbation of these. In this Colouring Topologies approach, we prove, for instance, that concise proofs to CPBT exist for instances contained within many infinite-sized sets of trees.