Signed permutations and the four color theorem

Let σ1,σ2σ1,σ2 be two permutations in the symmetric group SnSn. Among the many sequences of elementary transpositions τ1,…,τrτ1,…,τr transforming σ1σ1 into σ2=τr⋯τ1σ1σ2=τr⋯τ1σ1, 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≥2n≥2 and any σ1,σ2∈Snσ1,σ2∈Sn, there exists at least one signable sequence of elementary transpositions from σ1σ1 to σ2σ2. 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+2n+2 vertices by permutations in SnSn.