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# Partial Latin rectangle graphs and autoparatopism groups of partial Latin rectangles with trivial autotopism groups

Partial Latin rectangle graphs and autoparatopism groups of partial Latin rectangles with trivial autotopism groups
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 Author: Falcón Ganfornina, Raúl Manuel Stones, Rebecca J. Department: Universidad de Sevilla. Departamento de Matemática Aplicada I Date: 2017 Published in: Discrete Mathematics, 340 (6), 1242-1260. Document type: Article Abstract: An $r \times s$ partial Latin rectangle $(l_{ij})$ is an $r \times s$ matrix containing elements of $\{1,2,\ldots,n\} \cup \{\cdot\}$ such that each row and each column contain at most one copy of any symbol in $\{1,2,\ldots,n\}$. An entry is a triple $(i,j,l_{ij})$ with $l_{ij} \neq \cdot$. Partial Latin rectangles are operated on by permuting the rows, columns, and symbols, and by uniformly permuting the coordinates of the set of entries. The stabilizers under these operations are called the autotopism group and the autoparatopism group, respectively. We develop the theory of symmetries of partial Latin rectangles, introducing the concept of a partial Latin rectangle graph. We give constructions of $m$-entry partial Latin rectangles with trivial autotopism groups for all possible autoparatopism groups (up to isomorphism) when: (a) $r=s=n$, i.e., partial Latin squares, (b) $r=2$ and $s=n$, and (c) $r=2$ and $s \neq n$. Cite: Falcón Ganfornina, R.M. y Stones, R.J. (2017). Partial Latin rectangle graphs and autoparatopism groups of partial Latin rectangles with trivial autotopism groups. Discrete Mathematics, 340 (6), 1242-1260.
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DOI: 10.1016/j.disc.2017.01.002

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