Latin squares with maximal partial transversals of many lengths

Evans, Anthony B.; Mammoliti, Adam; Wanless, Ian M.

doi:10.1016/j.jcta.2021.105403

Cited by 3 publications

(3 citation statements)

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“…Then Evans [14] constructed an infinite family of Latin squares which simultaneously have maximal partial transversals of each of the permissible lengths. Subsequently, Evans et al [15] showed that there exists a Latin square of order n which has maximal partial transversals of each permissible length if and only if n {3, 4} ∉ and n 2 (mod 4) ≢ . In Latin squares, it is easy to find a partial transversal of length n 2 ⌈ ∕ ⌉ using a greedy algorithm.…”

Section: Every M Nmentioning

confidence: 99%

Small Latin arrays have a near transversal

Best

Pula

Wanless

2021

J of Combinatorial Designs

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Section: Every M Nmentioning

confidence: 99%

Small Latin arrays have a near transversal

Best

Pula

Wanless

2021

J of Combinatorial Designs

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“…. , n k ] that minimises n k 2 [11] 3 [11,17] 4 [11,17,28] 5 [11,17,31,41] 6 [11,17,28,46,58] 7 [11,17,28,42,64,78] 8 [11,17,28,42,63,90, 107] 9 [11,17,28,46,58,91,122,140] 10 [11,17,28,42,64,78,122,157,177] 11 [11,17,28,42,63,90,107,165,204,226] 12 [11,17,…”

Section: Long Partial Transversalsmentioning

confidence: 99%

“…Then Evans [16] constructed an infinite family of Latin squares which simultaneously have maximal partial transversals of each of the permissible lengths. Subsequently, Evans et al [17] showed that there exists a Latin square of order n which has maximal partial transversals of each permissible length if and only if n / ∈ {3, 4} and n ≡ 2 (mod 4). In Latin squares, it is easy to find a partial transversal of length ⌈n/2⌉ using a greedy algorithm.…”

Section: Introductionmentioning

confidence: 99%

Small Latin arrays have a near transversal

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Pula²,

Wanless³

2019

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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.

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