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Further results on latin squares with disjoint subsquares using rational outline squares

Tara Kemp, James Lefevre

TL;DR

This work develops a unified framework based on symmetric rational outline squares to address Latin squares with pairwise disjoint subsquares. By embedding realizability into the existence of symmetric rational outline squares (ROS) and their variants, the authors derive a single strengthened necessary condition that subsumes earlier criteria, and they provide constructive methods to obtain ROS for broad families of partitions, including cases with multiple equal subsquare sizes. Through the notions of similar outline squares (SROS) and equitable graph colouring, they translate ROS existence into explicit realizations (RP) via a frequency-array based construction, and they extend these ideas to conjectured families (largest-3 and $k-2$) and partitions with repeated parts. The results yield both new existence results and sharper necessary conditions, offering a promising route toward a general resolution of realizations for varied partition structures. Overall, the paper advances the theory of incomplete latin squares by linking outline-square techniques with constructive lifting to realizations and by showing that rational outline squares can capture and generalize many known and conjectured cases.

Abstract

In this paper we consider the problem of finding latin squares with sets of pairwise disjoint subsquares. We develop a new necessary condition on the sizes of the subsquares which incorporates and extends the known conditions. We provide a construction for the case where all but two of the subsquares are the same size, and in this case the condition is sufficient. We obtain these results using symmetric rational outline squares, and additionally provide several new results and extensions to this theory.

Further results on latin squares with disjoint subsquares using rational outline squares

TL;DR

This work develops a unified framework based on symmetric rational outline squares to address Latin squares with pairwise disjoint subsquares. By embedding realizability into the existence of symmetric rational outline squares (ROS) and their variants, the authors derive a single strengthened necessary condition that subsumes earlier criteria, and they provide constructive methods to obtain ROS for broad families of partitions, including cases with multiple equal subsquare sizes. Through the notions of similar outline squares (SROS) and equitable graph colouring, they translate ROS existence into explicit realizations (RP) via a frequency-array based construction, and they extend these ideas to conjectured families (largest-3 and ) and partitions with repeated parts. The results yield both new existence results and sharper necessary conditions, offering a promising route toward a general resolution of realizations for varied partition structures. Overall, the paper advances the theory of incomplete latin squares by linking outline-square techniques with constructive lifting to realizations and by showing that rational outline squares can capture and generalize many known and conjectured cases.

Abstract

In this paper we consider the problem of finding latin squares with sets of pairwise disjoint subsquares. We develop a new necessary condition on the sizes of the subsquares which incorporates and extends the known conditions. We provide a construction for the case where all but two of the subsquares are the same size, and in this case the condition is sufficient. We obtain these results using symmetric rational outline squares, and additionally provide several new results and extensions to this theory.
Paper Structure (12 sections, 35 theorems, 41 equations, 5 figures)

This paper contains 12 sections, 35 theorems, 41 equations, 5 figures.

Table of Contents

  1. Preliminaries
  2. Known results
  3. Outline squares
  4. Necessary conditions
  5. Rational solutions to two conjectures
  6. Similar outline squares
  7. Partitions with many parts of the same size
  8. Constructing realizations from rational outline squares
  9. Frequency arrays
  10. Equitable graph colouring
  11. Constructing the array $B$
  12. Concluding remarks

Key Result

Theorem 1.2

Take a partition $(h_1h_2\dots h_k)$ of $n$ with $h_1\geq h_2\geq \dots\geq h_k > 0$. Then an $\mathop{\mathrm{RP}}\nolimits(h_1h_2\dots h_k)$

Figures (5)

  • Figure 1: Latin squares with disjoint subsquares
  • Figure 2: Outline squares associated to $P = (3^12^3)$.
  • Figure 4: Structure of the outline array $O$
  • Figure 5: A latin square of order 6 with 4 disjoint transversals
  • Figure 6: The outline arrays $O_1$ when $m=3$ and $|A_1| = m$

Theorems & Definitions (65)

  • Example 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Conjecture 1.6
  • Conjecture 1.7
  • Theorem 1.8
  • Definition 2.1
  • Definition 2.2
  • ...and 55 more