Premium
Small Latin arrays have a near transversal
Author(s) -
Best Darcy,
Pula Kyle,
Wanless Ian M.
Publication year - 2021
Publication title -
journal of combinatorial designs
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.618
H-Index - 34
eISSN - 1520-6610
pISSN - 1063-8539
DOI - 10.1002/jcd.21778
Subject(s) - diagonal , transversal (combinatorics) , row , mathematics , latin square , combinatorics , row and column spaces , corollary , matrix (chemical analysis) , order (exchange) , diagonal matrix , computation , main diagonal , algorithm , geometry , mathematical analysis , computer science , rumen , chemistry , food science , materials science , finance , fermentation , economics , database , composite material
A Latin array is a matrix of symbols in which no symbol occurs more than once within a row or within a column. A diagonal of an n × n array is a selection of n cells taken from different rows and columns of the array. The weight of a diagonal is the number of different symbols on it. We show via computation that every Latin array of order n ⩽ 11 has a diagonal of weight at least n − 1 . A corollary is the existence of near transversals in Latin squares of these orders. More generally, for all k ⩽ 20 we compute a lower bound on the order of any Latin array that does not have a diagonal of weight at least n − k .