Article
Full entry |
PDF
(0.2 MB)
Feedback
PDF
(0.2 MB)
Feedback
Keywords:
latin square; latin trade; critical set
latin square; latin trade; critical set
Summary:
A latin trade is a subset of a latin square which may be replaced with a disjoint mate to obtain a new latin square. A $k$-homogeneous latin trade is one which intersects each row, each column and each entry of the latin square either $0$ or $k$ times. In this paper, we show that a construction given by Cavenagh, Donovan and Drápal for $3$-homogeneous latin trades in fact classifies every minimal $3$-homogeneous latin trade. We in turn classify all $3$-homogeneous latin trades. A corollary is that any $3$-homogeneous latin trade may be partitioned into three, disjoint, partial transversals.
References:
[1] Bate J.A., van Rees G.H.J.: Minimal and near-minimal critical sets in back circulant latin squares. Australas. J. Combin. 27 (2003), 47-61. MR 1955387 | Zbl 1024.05014
[2] Cavenagh N.J.: Embedding $3$-homogeneous latin trades into abelian $2$-groups. Comment. Math. Univ. Carolin. 45 (2004), 191-212. MR 2075269 | Zbl 1099.05503
[3] Cavenagh N.J.: The size of the smallest latin trade in a back circulant latin square. Bull. Inst. Combin. Appl. 38 (2003), 11-18. MR 1977014 | Zbl 1046.05015
[4] Cavenagh N.J., Donovan D., Drápal A.: $3$-homogeneous latin trades. Discrete Math. 300 (2005), 57-70. MR 2170114 | Zbl 1073.05012
[5] Cavenagh N.J., Donovan D., Drápal A.: $4$-homogeneous latin trades. Australas. J. Combin. 32 (2005), 285-303. MR 2139816 | Zbl 1074.05020
[6] Cooper J., Donovan D., Seberry J.: Latin squares and critical sets of minimal size. Australas. J. Combin. 4 (1991), 113-120. MR 1129273 | Zbl 0759.05017
[7] Donovan D., Howse A., Adams P.: A discussion of latin interchanges. J. Combin. Math. Combin. Comput. 23 (1997), 161-182. MR 1432756 | Zbl 0867.05010
[8] Donovan D., Mahmoodian E.S.: An algorithm for writing any latin interchange as the sum of intercalates. Bull. Inst. Combin. Appl. 34 (2002), 90-98. MR 1880972
[9] Drápal A.: On a planar construction of quasigroups. Czechoslovak Math. J. 41 (1991), 538-548. MR 1117806
[10] Drápal A.: Hamming distances of groups and quasi-groups. Discrete Math. 235 (2001), 189-197. MR 1829848 | Zbl 0986.20065
[11] Drápal A.: Geometry of latin trades. manuscript circulated at the conference Loops'03, Prague, 2003.
[12] Drápal A., Kepka T.: Exchangeable Groupoids I. Acta Univ. Carolin. Math. Phys. 24 (1983), 57-72. MR 0733686
[13] Drápal A., Kepka T.: On a distance of groups and latin squares. Comment. Math. Univ. Carolin. 30 (1989), 621-626. MR 1045889
[14] Fu C.-M., Fu H.-L.: The intersection problem of latin squares. J. Combin. Inform. System Sci. 15 (1990), 89-95. MR 1125351 | Zbl 0743.05009
[15] Horak P., Aldred R.E.L., Fleischner H.: Completing Latin squares: critical sets. J. Combin. Designs 10 (2002), 419-432. MR 1932121 | Zbl 1025.05011
[16] Keedwell A.D.: Critical sets for latin squares, graphs and block designs: A survey. Congr. Numer. 113 (1996), 231-245. MR 1393712 | Zbl 0955.05019
[17] Keedwell A.D.: Critical sets in latin squares and related matters: an update. Util. Math. 65 (2004), 97-131. MR 2048415 | Zbl 1053.05019
[18] Khosrovshahi G.B., Maimani H.R., Torabi R.: On trades: an update. Discrete Appl. Math. 95 (1999), 361-376. MR 1708848 | Zbl 0935.05015
[19] Street A.P.: Trades and defining sets. in: C.J. Colbourn and J.H. Dinitz, ed., CRC Handbook of Combinatorial Designs, CRC Press, Boca Raton, FL., USA, 1996, 474-478. Zbl 0847.05011
Search
Partner of
