On completing three cyclically generated transversals to a latin square

Cavenagh, Nicholas J., Hamalainen, Carlo and Nelson, Adrian M. (2009) On completing three cyclically generated transversals to a latin square. Finite Fields and Their Applications, 15 3: 294-303. doi:10.1016/j.ffa.2008.05.009

Author Cavenagh, Nicholas J.Hamalainen, CarloNelson, Adrian M. On completing three cyclically generated transversals to a latin square Finite Fields and Their Applications   Check publisher's open access policy 1071-57971090-2465 2009-06-01 Article (original research) 10.1016/j.ffa.2008.05.009 15 3 294 303 10 Maryland Heights, MO, United States Academic Press eng 2604 Applied Mathematics2602 Algebra and Number Theory2614 Theoretical Computer Science2200 Engineering Let P be a partial latin square of prime order p > 7 consisting of three cyclically generated transversals. Specifically, let P be a partial latin square of the form:P = {(i, c + i, s + i), (i, c + i, s + i), (i, c + i, s + i) | 0 ≤ i < p} for some distinct c, c, c and some distinct s, s, s. In this paper we show that any such P completes to a latin square which is diagonally cyclic. Equivalently, we prove that for each prime p > 7, every partial transversal of size 3 in the addition table for the integers modulo p can be completed to a full transversal. Let P be a partial latin square of prime order p > 7 consisting of three cyclically generated transversals. Specifically, let P be a partial latin square of the form:P = {(i, c + i, s + i), (i, c′ + i, s′ + i), (i, c″ + i, s″ + i) | 0 ≤ i < p} for some distinct c, c′, c″ and some distinct s, s′, s″. In this paper we show that any such P completes to a latin square which is diagonally cyclic. Equivalently, we prove that for each prime p > 7, every partial transversal of size 3 in the addition table for the integers modulo p can be completed to a full transversal. Latin squareDiagonally cyclic latin squareTransversalComplete mapping C1 Provisional Code UQ

 Document type: Journal Article Article (original research) School of Mathematics and Physics

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