Latin Squares whose transversals share many entries

2024/12/17 by Ghafari, Afsane, Wanless, Ian M. · 1 citation
#05B15 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2412.12466

Abstract

We prove that, for all even n≥10, there exists a latin square of order n with at least one transversal, yet all transversals coincide on \lfloor n/6 \rfloor entries. These latin squares have at least 19 n²/36 + O(n) transversal-free entries. We also prove that for all odd m≥ 3, there exists a latin square of order n=3m divided into nine m× m subsquares, where every transversal hits each of these subsquares at least once.

Cited by

• Entropy Bounds for Perfect Matchings in Bipartite Hypergraphs

Related

• Latin Squares whose transversals intersect in unusual ways
• Transversals in Latin Squares
• Quasi-transversal in Latin Squares
• Almost every Latin square has a decomposition into transversals
• Almost-full transversals in equi-n-squares