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Hamilton transversals in random Latin squares
Author(s) -
Gould Stephen,
Kelly Tom
Publication year - 2023
Publication title -
random structures and algorithms
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.314
H-Index - 69
eISSN - 1098-2418
pISSN - 1042-9832
DOI - 10.1002/rsa.21102
Subject(s) - latin square , transversal (combinatorics) , conjecture , mathematics , combinatorics , upper and lower bounds , square (algebra) , latin americans , least squares function approximation , statistics , law , mathematical analysis , geometry , chemistry , food science , fermentation , rumen , estimator , political science
Gyárfás and Sárközy conjectured that everyn × n $$ n\times n $$ Latin square has a “cycle‐free” partial transversal of size n − 2 $$ n-2 $$ . We confirm this conjecture in a strong sense for almost all Latin squares, by showing that asn → ∞ $$ n\to \infty $$ , all but a vanishing proportion ofn × n $$ n\times n $$ Latin squares have a Hamilton transversal, that is, a full transversal for which any proper subset is cycle‐free. In fact, we prove a counting result that in almost all Latin squares, the number of Hamilton transversals is essentially that of Taranenko's upper bound on the number of full transversals. This result strengthens a result of Kwan (which in turn implies that almost all Latin squares also satisfy the famous Ryser–Brualdi–Stein conjecture).