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Hamilton decompositions of the directed 3-torus: a return-map and odometer view

SangHyun Park

Abstract

We prove that the directed 3-torus D_3(m), or equivalently the Cartesian product of three directed m-cycles, admits a decomposition into three arc-disjoint directed Hamilton cycles for every integer m >= 3. The proof reduces Hamiltonicity to the m-step return maps on the layer section S=i+j+k=0. For odd m, five Kempe swaps of the canonical coloring produce return maps that are explicitly affine-conjugate to the standard 2-dimensional odometer. For even m, a sign-product invariant rules out Kempe-from-canonical constructions, and a different low-layer witness reduces after one further first-return map to a finite-defect clock-and-carry system. The remaining closure is a finite splice analysis, and the case m=4 is handled separately by a finite witness. A Lean 4 formalization accompanies the construction.

Hamilton decompositions of the directed 3-torus: a return-map and odometer view

Abstract

We prove that the directed 3-torus D_3(m), or equivalently the Cartesian product of three directed m-cycles, admits a decomposition into three arc-disjoint directed Hamilton cycles for every integer m >= 3. The proof reduces Hamiltonicity to the m-step return maps on the layer section S=i+j+k=0. For odd m, five Kempe swaps of the canonical coloring produce return maps that are explicitly affine-conjugate to the standard 2-dimensional odometer. For even m, a sign-product invariant rules out Kempe-from-canonical constructions, and a different low-layer witness reduces after one further first-return map to a finite-defect clock-and-carry system. The remaining closure is a finite splice analysis, and the case m=4 is handled separately by a finite witness. A Lean 4 formalization accompanies the construction.

Paper Structure

This paper contains 47 sections, 44 theorems, 259 equations, 1 figure, 7 tables.

Table of Contents

  1. Introduction
  2. Contemporaneous and independent work.
  3. Colorings, Kempe swaps, and return maps
  4. Arc-colorings and direction assignments
  5. Kempe maps and Kempe swaps
  6. A parity barrier
  7. The layer function and return maps
  8. Odd m: an explicit affine odometer witness
  9. The affine supports
  10. Even m: repairing the clock with Route E
  11. Layers $S\notin\{0,1,2\}$.
  12. Layer $S=1$.
  13. Layer $S=2$.
  14. Layer $S=0$, case $m\equiv 0,2\pmod 6$.
  15. Layer $S=0$, case $m\equiv 4\pmod 6$.
  16. ...and 32 more sections

Key Result

Theorem 1

For every integer $m\ge 3$, the arc set of $D_3(m)$ decomposes into three directed Hamilton cycles.

Figures (1)

  • Figure 1: Schematic defect geometry on $P_0$. The left and right panels use the bulk coordinates of Lemma \ref{['lem:routeE-bulkcoords']}; the center panel (color $2$) uses the working frame $(x,y)=(i,\,i+k)$. In each panel the arrows indicate the generic bulk branch, while the solid/dashed/dotted lines represent the affine defect families from Lemma \ref{['lem:routeE-finite-defect']}. The isolated boundary-point corrections sit on these lines and act as the splice points described in Remark \ref{['rem:routeE-design']}.

Theorems & Definitions (107)

  • Theorem 1
  • Example 1: Canonical coloring
  • Definition 1: Direction assignment
  • Definition 2: Kempe swap
  • Lemma 1: Kempe swaps preserve validity
  • proof
  • Theorem 2: Sign-product invariant
  • proof
  • Corollary 1: Parity barrier for Kempe-from-canonical when $m$ is even
  • proof
  • ...and 97 more