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On resolvable designs $$S_ 3(3;4,v)$$. (English) Zbl 0648.05007
We study a method of Lonz and Vanstone which constructs an $$S_ 3(3,4,2n)$$ from any given 1-factorization of $$K_{2n}$$. We show that the resulting designs admit at least 3 mutually orthogonal resolutions whenever $$n\geq 4$$ is even. In particular, the necessary conditions for the existence of a resolvable $$S_ 3(3,4,v)$$ are also sufficient. Examples without repeated blocks are shown to exist provided that $$n\not\equiv 2$$ mod 3.

##### MSC:
 05B05 Combinatorial aspects of block designs 05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.)
##### Keywords:
resolvability; designs; mutually orthogonal resolutions
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##### References:
 [1] Beth, Th; Jungnickel, D, Einige einfache fahnenhomogene 3-blockpläne, Math. Z., 183, 443-445, (1983) · Zbl 0531.05010 [2] Beth, Th; Jungnickel, D; Lenz, H, Design theory, (1985), Bibliographisches Institut Mannheim/Wien/Zürich [3] Dinitz, J.H; Stinson, D.R, The spectrum of room cubes, European J. combin, 2, 221-230, (1981) · Zbl 0531.05014 [4] Gross, K.B; Mullin, R.C; Wallis, W.D, The number of pairwise orthogonal symmetric Latin squares, Utilitas math., 4, 239-251, (1973) · Zbl 0273.05011 [5] Hanani, H, On some tactical configurations, Canad. J. math., 15, 705-722, (1963) · Zbl 0196.29102 [6] Hartman, A; Rosa, A, Cyclic one-factorization of the complete graph, European J. combin., 6, 45-48, (1985) · Zbl 0624.05051 [7] Horton, J.D, Room designs and 1-factorizations, Aequationes math., 22, 56-63, (1981) · Zbl 0466.05012 [8] \scD. Jungnickel and S. A. Vanstone, Hyperfactorizations of graphs and 5-designs, to appear. · Zbl 0639.05039 [9] Köhler, E, k-difference cycles and the construction of cyclic t-designs, (), 195-203 [10] \scS. A. Lonz and S. A. Vanstone, Private communication.
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