# Counting Locally Flat-Foldable Origami Configurations Via 3-Coloring Graphs

## Abstract

Origami, where two-dimensional sheets are folded into complex structures, is rich with combinatorial and geometric structure, most of which remains to be fully understood. In this paper we consider flat origami, where the sheet of material is folded into a two-dimensional object, and consider the mountain (convex) and valley (concave) creases that result, called a MV assignment of the crease pattern. An open problem is to count the number locally valid MV assignments $$\mu$$ of a given flat-foldable crease pattern C, where locally valid means that each vertex will fold flat under $$\mu$$ with no self-intersections of the folded material. In this paper we solve this problem for a large family of crease patterns by creating a planar graph $$C^*$$ whose 3-colorings are in one-to-one correspondence with the locally valid MV assignments of C. This reduces the problem of enumerating locally valid MV assignments to the enumeration of 3-colorings of graphs.

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## Acknowledgements

This material is based upon work supported by the National Science Foundation under Grant number DMS-1851842, the MathILy-EST Research Experience for Undergraduates, under the direction in 2019 of the third author. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. The authors wish to thank Bryn Mawr College for providing excellent hospitality and facilities for conducting this work.

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Correspondence to Thomas C. Hull.

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Chiu, A., Hoganson, W., Hull, T.C. et al. Counting Locally Flat-Foldable Origami Configurations Via 3-Coloring Graphs. Graphs and Combinatorics 37, 241–261 (2021). https://doi.org/10.1007/s00373-020-02240-2

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• DOI: https://doi.org/10.1007/s00373-020-02240-2

### Keywords

• Graph coloring
• Origami

• 05C15
• 52C99