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doi:10.1016/0304-3975(95)00126-3 
Copyright © 1996 Published by Elsevier Science B.V.
Guthrie's problem: new equivalences and rapid reductions
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Available online 10 February 1999.
Abstract
In 1977, Appel and Haken proved that every planar graph is four vertex colourable which finally proved Guthrie's conjecture of circa 1852 that four colours are always sufficient. Their proof is very long and the implicit algorithm for four colouring is rather impractical. This paper provides a new characterisation of the four-colour problem by showing that it is equivalent (by an optimally fast reduction) to a simply stated problem of 3-edge colouring pairs of trees. This new problem, in turn, is equivalent to nontrivial subclasses of other problems in mathematics and computer science of which we describe three. These are problems of intersection of regular languages, of integer linear equations and of algebraic expressions. In the general case, all these problems require exponential time to solve. We show that if these problems are defined on pairs of trees, then polynomial time is sufficient. In addition, these problems offer enticing opportunities in the search for a shorter proof of the four-colour theorem and for more practical algorithms for four-colouring planar graphs.
