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• W2775536822 abstract "We define a cover of a Latin square to be a set of entries that includes at least one representative of each row, column and symbol. A cover is minimal if it does not contain any smaller cover. A partial transversal is a set of entries that includes at most one representative of each row, column and symbol. A partial transversal is maximal if it is not contained in any larger partial transversal. We explore the relationship between covers and partial transversals. We prove the following: (1) The minimum size of a cover in a Latin square of order $n$ is $n+a$ if and only if the maximum size of a partial transversal is either $n-2a$ or $n-2a+1$. (2) A minimal cover in a Latin square of order $n$ has size at most $mu_n=3(n+1/2-sqrt{n+1/4})$. (3) There are infinitely many orders $n$ for which there exists a Latin square having a minimal cover of every size from $n$ to $mu_n$. (4) Every Latin square of order $n$ has a minimal cover of a size which is asymptotically equal to $mu_n$. (5) If $1le kle n/2$ and $nge5$ then there is a Latin square of order $n$ with a maximal partial transversal of size $n-k$. (6) For any $epsilon>0$, asymptotically almost all Latin squares have no maximal partial transversal of size less than $n-n^{2/3+epsilon}$." @default.
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• W2775536822 date "2017-12-07" @default.
• W2775536822 modified "2023-09-25" @default.
• W2775536822 title "Covers and partial transversals of Latin squares" @default.
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