On signed diagonal flip sequences

TL;DR

This paper investigates the conditions under which a sequence of diagonal flips in polygon triangulations can be signed, linking combinatorial transformations to the Four Color Theorem.

Contribution

It provides a necessary and sufficient condition for a diagonal flip sequence to be a signed diagonal flip sequence, advancing understanding of polygon triangulation transformations.

Findings

Established a criterion for signed diagonal flip sequences

Connected flip sequences to the Four Color Theorem

Enhanced understanding of triangulation transformations

Abstract

Eliahou \cite{2} and Kryuchkov \cite{9} conjectured a proposition that Gravier and Payan \cite{4} proved to be equivalent to the Four Color Theorem. It states that any triangulation of a polygon can be transformed into another triangulation of the same polygon by a sequence of signed diagonal flips. It is well known that any pair of polygonal triangulations are connected by a sequence of (non-signed) diagonal flips. In this paper we give a sufficient and necessary condition for a diagonal flip sequence to be a signed diagonal flip sequence.