Counting locally flat-foldable origami configurations via 3-coloring graphs

TL;DR

This paper introduces a method to count locally flat-foldable origami configurations by transforming crease patterns into planar graphs, where 3-colorings correspond to valid foldings, simplifying the enumeration process.

Contribution

It establishes a novel approach linking flat origami MV assignments to 3-colorings of planar graphs, enabling easier enumeration of foldable configurations.

Findings

Creates a planar graph $C^*$ from crease pattern $C$

3-colorings of $C^*$ correspond to valid MV assignments

Reduces origami folding enumeration to graph coloring problem

Abstract

Origami, where two-dimensional sheets are folded into complex structures, is proving to be rich with combinatorial and geometric structure, most of which remains to be fully understood. In this paper we consider \emph{flat origami}, where the sheet of material is folded into a two-dimensional object, and consider the mountain (convex) and valley (concave) creases made by such foldings, called a \emph{MV assignment} of the crease pattern. We establish a method to, given a flat-foldable crease pattern $C$ under certain conditions, create a planar graph $C^*$ whose 3-colorings are in one-to-one correspondence with the locally-valid MV assignments of $C$. This reduces the general, unsolved problem of enumerating locally-valid MV assignments to the enumeration of 3-colorings of graphs.