| Title:
|
A uniqueness result for $3$-homogeneous latin trades (English) |
| Author:
|
Cavenagh, Nicholas J. |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
47 |
| Issue:
|
2 |
| Year:
|
2006 |
| Pages:
|
337-358 |
| . |
| Category:
|
math |
| . |
| 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. (English) |
| Keyword:
|
latin square |
| Keyword:
|
latin trade |
| Keyword:
|
critical set |
| MSC:
|
05B15 |
| idZBL:
|
Zbl 1138.05007 |
| idMR:
|
MR2241536 |
| . |
| Date available:
|
2009-05-05T16:57:42Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119596 |
| . |
| Reference:
|
[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. Zbl 1024.05014, MR 1955387 |
| Reference:
|
[2] Cavenagh N.J.: Embedding $3$-homogeneous latin trades into abelian $2$-groups.Comment. Math. Univ. Carolin. 45 (2004), 191-212. Zbl 1099.05503, MR 2075269 |
| Reference:
|
[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. Zbl 1046.05015, MR 1977014 |
| Reference:
|
[4] Cavenagh N.J., Donovan D., Drápal A.: $3$-homogeneous latin trades.Discrete Math. 300 (2005), 57-70. Zbl 1073.05012, MR 2170114 |
| Reference:
|
[5] Cavenagh N.J., Donovan D., Drápal A.: $4$-homogeneous latin trades.Australas. J. Combin. 32 (2005), 285-303. Zbl 1074.05020, MR 2139816 |
| Reference:
|
[6] Cooper J., Donovan D., Seberry J.: Latin squares and critical sets of minimal size.Australas. J. Combin. 4 (1991), 113-120. Zbl 0759.05017, MR 1129273 |
| Reference:
|
[7] Donovan D., Howse A., Adams P.: A discussion of latin interchanges.J. Combin. Math. Combin. Comput. 23 (1997), 161-182. Zbl 0867.05010, MR 1432756 |
| Reference:
|
[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 |
| Reference:
|
[9] Drápal A.: On a planar construction of quasigroups.Czechoslovak Math. J. 41 (1991), 538-548. MR 1117806 |
| Reference:
|
[10] Drápal A.: Hamming distances of groups and quasi-groups.Discrete Math. 235 (2001), 189-197. Zbl 0986.20065, MR 1829848 |
| Reference:
|
[11] Drápal A.: Geometry of latin trades.manuscript circulated at the conference Loops'03, Prague, 2003. |
| Reference:
|
[12] Drápal A., Kepka T.: Exchangeable Groupoids I.Acta Univ. Carolin. Math. Phys. 24 (1983), 57-72. MR 0733686 |
| Reference:
|
[13] Drápal A., Kepka T.: On a distance of groups and latin squares.Comment. Math. Univ. Carolin. 30 (1989), 621-626. MR 1045889 |
| Reference:
|
[14] Fu C.-M., Fu H.-L.: The intersection problem of latin squares.J. Combin. Inform. System Sci. 15 (1990), 89-95. Zbl 0743.05009, MR 1125351 |
| Reference:
|
[15] Horak P., Aldred R.E.L., Fleischner H.: Completing Latin squares: critical sets.J. Combin. Designs 10 (2002), 419-432. Zbl 1025.05011, MR 1932121 |
| Reference:
|
[16] Keedwell A.D.: Critical sets for latin squares, graphs and block designs: A survey.Congr. Numer. 113 (1996), 231-245. Zbl 0955.05019, MR 1393712 |
| Reference:
|
[17] Keedwell A.D.: Critical sets in latin squares and related matters: an update.Util. Math. 65 (2004), 97-131. Zbl 1053.05019, MR 2048415 |
| Reference:
|
[18] Khosrovshahi G.B., Maimani H.R., Torabi R.: On trades: an update.Discrete Appl. Math. 95 (1999), 361-376. Zbl 0935.05015, MR 1708848 |
| Reference:
|
[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 |
| . |
