Abstract
In this paper, we construct Room frames with partitionable transversals. Direct and recursive constructions are used to find sets of disjoint complete ordered partitionable (COP) transversals and sets of disjoint holey ordered partitionable (HOP) transversals for Room frames. Our main results include upper and lower bounds on the number of disjoint COP transversals and the number of disjoint HOP transversals for Room frames of type 2n. This work is motivated by the large number of applications of these designs.
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R. J. R. Abel, A. E. Brouwer, C. J. Colbourn, and J. H. Dinitz, Mutually orthogonal Latin squares (MOLS), CRC Handbook of Combinatorial Design (C. J. Colbourn and J. H. Dinitz, eds.), CRC Press, Boca Raton, FL (1996) pp. 111–142.
R. J. R. Abel, C. J. Colbourn, and J. H. Dinitz, Incomplete MOLS, CRC Handbook of Combinatorial Designs (C. J. Colbourn and J. H. Dinitz, eds.), CRC Press, Boca Raton, FL (1996) pp. 142–172.
J. H. Dinitz and D. K Garnick, Holey factorizations, Ars Combin., 44 (1996) pp. 65–92.
J. H. Dinitz and E. R. Lamken, Howell designs with sub-designs, J. Combin. Theory (A), Vol. 65 (1994) pp. 268–301.
J. H. Dinitz and E. R. Lamken, Uniform Room frames with five holes, J. of Combinatorial Design, Vol. 1 (1993) pp. 323–328.
J. H. Dinitz and D. R. Stinson, A few more Room frames, Graphs, Matrices and Designs, Marcel Dekker, New York (1993) pp. 133–146.
J. H. Dinitz and D. R. Stinson, The construction and uses of frames, Ars Combin., Vol. 10 (1980) pp. 31–54.
J. H. Dinitz and D. R. Stinson, A hill-climbing algorithm for the construction of one-factorizations and Room squares, SIAM J. Algebraic Discrete Methods, Vol. 8 (1987) pp. 430–438.
J. H. Dinitz, D. R. Stinson, and Zhu Lie, On the spectra of certain classes of Room frames, Electronic J. of Combinatorics, Vol. 1 (1994) #R7.
G. Ge and L. Zhu, On the existence of Room frames of types t u, J. of Combinatorial Designs, Vol. 1 (1993) pp. 183–191.
E. R. Lamken, The existence of 3 orthogonal partitioned incomplete Latin squares of type t n, Discrete Math., Vol. 89 (1991) pp. 231–251.
E. R. Lamken, The existence of partitioned generalized balanced tournament designs with block size 3, Designs, Codes and Cryptography, 11 (1997), pp. 37–71.
E. R. Lamken and S. A. Vanstone, The existence of skew Howell designs of side 2n and order 2n + 2, J. Combin. Theory (A), Vol. 54 (1990) pp. 20–40.
E. R. Lamken and S. A. Vanstone, Skew transversals in frames, J. Combin. Math. and Combin. Computing, Vol. 2 (1987) pp. 37–50.
C. C. Lindner, C. A. Rodger, and D. R. Stinson, Nesting of cycle systems of odd length, Discrete Math., Vol. 77 (1989) pp. 191–203.
R. C. Mullin, P. J. Schellenberg, S. A. Vanstone, and W. D. Wallis, On the existence of frames, Discrete Math., Vol. 37 (1981) pp. 79–104.
D. R. Stinson, On the existence of skew Room frames of type 2n, Ars Combin., Vol. 24 (1987) pp. 115–128.
D. R. Stinson, Some Classes of Frames, and the Spectra of Skew Room Frames and Howell Designs, Ph.D. Dissertation, University of Waterloo (1981).
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Dinitz, J.H., Lamken, E.R. HOPs and COPs: Room frames with partitionable transversals. Designs, Codes and Cryptography 19, 5–26 (2000). https://doi.org/10.1023/A:1008334412968
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DOI: https://doi.org/10.1023/A:1008334412968