Signed permutations and the four color theorem

Abstract

Let σ 1 , σ 2 be two permutations in the symmetric group S n . Among the many sequences of elementary transpositions τ 1 , … , τ r transforming σ 1 into σ 2 = τ r ⋯ τ 1 σ 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 σ 1 , σ 2 ∈ S n , there exists at least one signable sequence of elementary transpositions from σ 1 to σ 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 .