A property of trivalent graphs related to equidissections

TL;DR

This paper proves Stein's conjecture on polygon dissections into equal-area triangles under rational coordinate assumptions, using properties of trivalent graphs with rational flows.

Contribution

It introduces a new approach linking polygon dissection problems to properties of trivalent graphs with rational flows, proving Stein's conjecture under specific conditions.

Findings

Stein's conjecture holds for polygons with rational vertices.

A connection between polygon dissections and graph flow properties is established.

The proof relies on a general property of trivalent graphs with rational flows.

Abstract

Monsky proved that a square cannot be dissected into an odd number of triangles of equal area. Stein conjectured that the same holds for any polygon whose edges can be paired into parallel and equal-length segments. We prove Stein's conjecture under an assumption that all triangle vertices have rational coordinates. Our result is derived from a more general property of trivalent graphs equipped with a $\mathbb{Q}^2$-valued flow.