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A property of trivalent graphs related to equidissections

Daniil Rudenko

TL;DR

This work establishes a parity phenomenon for vertex weights in trivalent graphs carrying a $\mathbb{Q}^2$-valued flow: the count of vertices attaining the minimal weight $m(\Gamma,f)$ is even. The weights are defined via the $2$-adic valuation of the area form applied to flow vectors, linking graph-theoretic structure to equidissection problems. The authors prove the main theorem by reducing to integral flows using $2$-adic localization, analyzing lattices $L(v)$ at vertices, and exploiting odd cycles to construct a contradiction unless the parity condition holds. As a corollary, they derive a rational-version of Stein's conjecture: a special polygon whose vertices are rational cannot be dissected into an odd number of equal-area triangles. The framework connects Monsky-type nonexistence results for equidissections to a general parity property of $\mathbb{Q}^2$-flows on trivalent graphs, suggesting broad applicability under rationality assumptions.

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.

A property of trivalent graphs related to equidissections

TL;DR

This work establishes a parity phenomenon for vertex weights in trivalent graphs carrying a -valued flow: the count of vertices attaining the minimal weight is even. The weights are defined via the -adic valuation of the area form applied to flow vectors, linking graph-theoretic structure to equidissection problems. The authors prove the main theorem by reducing to integral flows using -adic localization, analyzing lattices at vertices, and exploiting odd cycles to construct a contradiction unless the parity condition holds. As a corollary, they derive a rational-version of Stein's conjecture: a special polygon whose vertices are rational cannot be dissected into an odd number of equal-area triangles. The framework connects Monsky-type nonexistence results for equidissections to a general parity property of -flows on trivalent graphs, suggesting broad applicability under rationality assumptions.

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 -valued flow.

Paper Structure

This paper contains 3 sections, 5 theorems, 2 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['MainTheorem']}
  3. Proof of Theorem \ref{['MainCorollary']}

Key Result

Theorem 1

Let $\Gamma$ be a trivalent graph with a flow $f\colon \vec{E} \to \mathbb{Q}^2$. Then the number of vertices $v \in V$ such that $m(v) = m(\Gamma)$ is even.

Theorems & Definitions (7)

  • Theorem 1
  • Theorem 2: Stein's conjecture for rational dissections
  • Lemma 3
  • Lemma 4
  • proof
  • Lemma 5
  • proof