Signed permutations and the four color theorem

TL;DR

This paper links permutations to polygon triangulations and uses a signed permutation framework to reformulate the four color theorem via paths in the Cayley graph of the symmetric group.

Contribution

It introduces a signed permutation construction that connects permutation classes with polygon triangulations and offers a new formulation of the four color theorem.

Findings

Fibers of permutation-triangulation association match sylvester classes

Signed permutations enable reformulation of the four color theorem

Path existence in Cayley graph relates to four colorability

Abstract

To each permutation $\sigma$ in $S_{n}$ we associate a triangulation of a fixed $(n+2)$-gon. We then determine the fibers of this association and show that they coincide with the sylvester classes depicted By Novelli, Hivert and Thibon. A signed version of this construction allows us to reformulate the four color theorem in terms of the existence of a signable path between any two permutations in the Cayley graph of the symmetric group $S_{n}.