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On the completion of latin rectangles to symmetric latin squares

  • Darryn Bryant (a1) and C. A. Rodger (a2)

Abstract

We find necessary and sufficient conditions for completing an arbitrary 2 by n latin rectangle to an n by n symmetric latin square, for completing an arbitrary 2 by n latin rectangle to an n by n unipotent symmetric latin square, and for completing an arbitrary 1 by n latin rectangle to an n by n idempotent symmetric latin square. Equivalently, we prove necessary and sufficient conditions for the existence of an (n−1)-edge colouring of Kn (n even), and for n-edge colouring of Kn (n odd) in which the colours assigned to the edges incident with two vertices are specified in advance.

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References

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[1]Andersen, L. D., ‘Embedding latin squares with prescribed diagonal’, in: Algebraic and geometric combinatorics, North-Holland Math. Stud. 65 (North-Holland, Amsterdam, 1982) pp. 926.
[2]Colbourn, C. J. and Dinitz, J. H., CRC handbook of combinatorial designs (CRC Press, Boca Raton FL, 1996).
[3]Cruse, A. B., ‘On embedding incomplete symmetric latin squares’, J. Combin. Theory Ser. A 16 (1974), 1822.
[4]Hall, P., ‘On representation of subsets’, J. London Math. Soc. 10 (1935), 2630.
[5]Rodger, C. A., ‘Embedding incomplete idempotent latin squares’, in: Combinatorial mathematics, X (Adelaide, 1982), Lecture Notes in Math. 1036 (Springer, Berlin, 1983) pp. 355366.
[6]Rodger, C. A., ‘Embedding an incomplete latin square in a latin square with a prescribed diagonal’, Discrete Math. (1) 51 (1984), 7389.
[7]Smetaniuk, B., ‘A new construction on latin squares - I: A proof of the Evans conjecture’, Ars Combin. 11 (1981), 155172.
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On the completion of latin rectangles to symmetric latin squares

  • Darryn Bryant (a1) and C. A. Rodger (a2)

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