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Sunlet factors for Cartesian products of cycles

Henry Jervis, Paul C. Kainen

TL;DR

The paper addresses factoring the toroidal grid $C_{2n} \Box C_{2n}$ into edge-disjoint sunlets by constructing 2-to-1 coverings from disjoint unions of sunlets of sizes $s \in \{1, n, n^2\}$, for $n \ge 3$. It develops explicit, coordinate-based coverings that preserve the sunlet structure and orientation, including compatibility with finite-state-machine (FSM) orientations. The main contributions are three theorems corresponding to $s=1$, $s=n$, and $s=n^2$, plus a minimum-bipartite-sunlet factorization, all yielding Hamiltonian-cycle-structured decompositions and staircase or odd-square patterns. This work demonstrates multiple rearrangements of a toroidal grid into sunlets with potential relevance for parallel computation on regular networks and oriented graph decompositions.

Abstract

A sunlet is a cycle with a pendant edge attached at each vertex of the cycle. For the bipartite toroidal grid graphs $C_{2n} \Box C_{2n}$, factorizations into sunlets are given by homomorphisms from disjoint unions of $s$ copies of a sunlet for $s \in \{1, n, n^2\}, n \geq 3$ such that edges are mapped bijectively.

Sunlet factors for Cartesian products of cycles

TL;DR

The paper addresses factoring the toroidal grid into edge-disjoint sunlets by constructing 2-to-1 coverings from disjoint unions of sunlets of sizes , for . It develops explicit, coordinate-based coverings that preserve the sunlet structure and orientation, including compatibility with finite-state-machine (FSM) orientations. The main contributions are three theorems corresponding to , , and , plus a minimum-bipartite-sunlet factorization, all yielding Hamiltonian-cycle-structured decompositions and staircase or odd-square patterns. This work demonstrates multiple rearrangements of a toroidal grid into sunlets with potential relevance for parallel computation on regular networks and oriented graph decompositions.

Abstract

A sunlet is a cycle with a pendant edge attached at each vertex of the cycle. For the bipartite toroidal grid graphs , factorizations into sunlets are given by homomorphisms from disjoint unions of copies of a sunlet for such that edges are mapped bijectively.
Paper Structure (6 sections, 3 theorems, 7 equations, 6 figures)

This paper contains 6 sections, 3 theorems, 7 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Definitions
  3. Homomorphism from One Sunlet
  4. Homomorphism from a Set of $n$ Sunlets
  5. Factorization into Minimum Bipartite Sunlets
  6. Conclusion

Key Result

Theorem 3.1

For $n \geq 3$, there is a 2-to-1 covering such that $\varphi(Z(S))$ is a Hamiltonian cycle. Further, $\varphi$ is compatible with the FSM orientation of $S$ and the standard orientation for the grid.

Figures (6)

  • Figure 1: $C_{4} \ \square \ C_{4}\;$ and $\;C_{3} \ \square \ C_{3}\;$ with described coordinates.
  • Figure 2: $C_{4} \ \square \ C_{4}$ with Homomorphism and Sample Vertex Types. The sunlet's cycle is in orange and its rays are in blue.
  • Figure 3: Visualized Point Subsets.
  • Figure 4: Homomorphism of $2 \cdot S_{8}^{1}$ onto $C_{4} \, \Box \, C_{4}$ on a toroidal grid. One $S_{8}^{1}$ is a red cycle with purple rays, the other is a green cycle with blue rays; opposite sides identified.
  • Figure 5: Covering of $4 \cdot S_{4}^{1}$ onto $C_{4}\, \Box\, C_{4}$ in blue, red, green, and yellow. The cycle edges are denoted by solid arrows and rays by shaded arrows which also indicate directionality.
  • ...and 1 more figures

Theorems & Definitions (6)

  • Theorem 3.1
  • proof
  • Theorem 4.1
  • proof
  • Theorem 5.1
  • proof