Do K33-Free Latin Squares Exist?

TL;DR

The paper investigates the existence of $K_{3,3}$-free latin squares, framing the problem through latin square graphs, transversal trades, and eigenfunction theory. It confirms extreme rarity for small orders via exhaustive search, reproduces the known order-$8$ examples, and introduces a switched construction yielding a $K_{3,3}$-free square of order $16$ from two orthogonal $8\times8$ squares. It also extends the analysis to orthogonal latin squares, establishing the existence of both $K_{4,4}$-free and non-$K_{4,4}$-free linear pairs for every odd prime-power order larger than $5$, using algebraic (GF$(q)$) constructions and pattern-type classifications. Together, these results connect pattern-avoidance in latin square graphs with transversal trades, eigenfunction supports, and switch-based constructions, while leaving open the broader existence problem for larger orders and generalizations to larger MOLS sets.

Abstract

We discuss the problem of existence of latin squares without a substructure consisting of six elements $(r_1,c_2,l_3)$, $(r_2,c_3,l_1)$, $(r_3,c_1,l_2)$, $(r_2,c_1,l_3)$, $(r_3,c_2,l_1)$, $(r_1,c_3,l_2)$. Equivalently, the corresponding latin square graph does not have an induced subgraph isomorphic to $K_{3,3}$. The exhaustive search [Brouwer, Wanless. Universally noncommutative loops. 2011] says that there are no such latin squares of order $3$--$7$, $9$--$11$ and there are only two $K_{3,3}$-free latin squares of order $8$, up to equivalence. We repeat the search, establishing also the number of latin $m$-by-$n$ rectangles for each $m$ and $n$ less than or equal to $11$. As a switched combination of two orthogonal latin squares of order $8$, we construct a $K_{3,3}$-free (universally noncommutative) latin square of order $16$. We also consider a similar problem for orthogonal latin squares, proving that there are both $K_{4,4}$-free and non-$K_{4,4}$-free linear pairs of orthogonal latin squares for each odd prime-power order larger than~$5$. Keywords: latin square; transversal; trade; pattern avoiding; eigenfunction; universally noncommutative loop.

Figures (7)

• Figure 1: The two $K_{3,3}$-free $8\times 8$ latin squares and related orthogonal $4\times 4$ latin squares
• Figure 2: The $K_{3,3}$-free latin rectangles $4\times 5$ and $8\times 9$, with their continuations to latin squares.
• Figure 3: Two $K_{3,3}$-free latin $11\times 12$ rectangles and one $K_{3,3}$-free latin $9\times 12$ rectangles without intercalates.
• Figure 4: The two $K_{3,3}$-free $8\times 8$ latin squares and related orthogonal $4\times 4$ latin squares
• Figure 5: Two orthogonal latin squares and a switching function that give a $K_{3,3}$-free latin square of order $16$ as a switched combitation

Theorems & Definitions (13)

• Proposition 1
• Proposition 2
• proof
• Proposition 3
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof