Latin Squares whose transversals share many entries

TL;DR

This paper constructs specific Latin squares of even and odd orders with many entries shared among all transversals, revealing new structural properties and bounds related to transversals in Latin squares.

Contribution

It introduces constructions of Latin squares where all transversals share many entries and establishes bounds on transversal-free entries for even orders.

Findings

Existence of Latin squares with transversals sharing at least loor(n/6)ixed entries for even n

Latin squares with at least 19n^2/36 + O(n) transversal-free entries for even n

Construction of Latin squares of order 3m with transversals hitting each subsquare at least once for odd m

Abstract

We prove that, for all even $n\geq10$, there exists a latin square of order $n$ with at least one transversal, yet all transversals coincide on $ \big\lfloor n/6 \big\rfloor$ entries. These latin squares have at least $ 19 n^2/36 + O(n)$ transversal-free entries. We also prove that for all odd $m\geq 3$, there exists a latin square of order $n=3m$ divided into nine $m\times m$ subsquares, where every transversal hits each of these subsquares at least once.