Signed permutations and the four color theorem

Let sigma(1), sigma(2) be two permutations in the symmetric group S-n. Among the many sequences of elementary transpositions tau(1), ... , tau(r) transforming sigma(1) into sigma(2) = tau(r) ... tau(1)sigma(1), some of them may be signable, a property introduced in this paper. We show that the four color theorem in graph theory is equivalent to the statement that, for any n >= 2 and any sigma(1), sigma(2) is an element of S-n, there exists at least one signable sequence of elementary transpositions from sigma(1) to sigma(2). This algebraic reformulation rests on a former geometric one in terms of signed diagonal flips, together with a codification of the triangulations of a convex polygon on n + 2 vertices by permutations in S-n. (C) 2009 Elsevier GmbH. All fights reserved.