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Permutation representation of 3 and 4-homogeneous Latin bitrades. (English) Zbl 1188.05039
Summary: Cavenagh, Donovan and Drápal (2005) [N. Cavenagh, D. Donovan, and A. Drápal, “3-homogeneous Latin trades”, Discrete Math. 300, No. 1-3, 57–70 (2005; Zbl 1073.05012)] gave a construction for 3-homogeneous latin bitrades which was proved by Cavenagh (2006) [N. Cavenagh, “A uniqueness result for 3-homogeneous Latin trades”, Commentat. Math. Univ. Carol. 47, No. 2, 337–358 (2006; Zbl 1111.05085)] to give every 3-homogeneous latin bitrade. A corollary is that every 3-homogeneous Latin bitrade may be partitioned into three disjoint transversals. Using a permutation representation for latin bitrades developed by Drápal, we provide a shorter proof of these results, and then give a construction for a family of separated 4-homogeneous latin bitrades.
05B15 Orthogonal arrays, Latin squares, Room squares