About DML-CZ | FAQ | Conditions of Use | Math Archives | Contact Us
  Previous |  Up |  Next

Article

Cavenagh, Nicholas J.
Embedding $3$-homogeneous latin trades into abelian $2$-groups. (English). Commentationes Mathematicae Universitatis Carolinae, vol. 45 (2004), issue 2, pp. 191-212
MSC: 05B15, 20N05 | MR 2075269 | Zbl 1099.05503
Keywords:
latin square; latin trade; abelian $2$-group
Summary:
Let $T$ be a partial latin square and $L$ be a latin square with $T\subseteq L$. We say that $T$ is a latin trade if there exists a partial latin square $T'$ with $T'\cap T=\emptyset $ such that $(L\setminus T)\cup T'$ is a latin square. A $k$-homogeneous latin trade is one which intersects each row, each column and each entry either $0$ or $k$ times. In this paper, we show the existence of $3$-homogeneous latin trades in abelian $2$-groups.
References:
[1] Adams P., Bean R., Khodkar A.: A census of critical sets in the latin squares of order at most six. Ars Combin. 68 (2003), 203-223. MR 1991049 | Zbl 1072.05511
[2] Adams P., Khodkar A.: Smallest critical sets for the latin squares of order six and seven. J. Combin. Math. Combin. Computing. 67 (2001), 225-237. MR 1834445
[3] Bates J.A., van Rees G.H.J.: The size of the smallest strong critical set in a latin square. Ars Combin. 53 (1999), 73-83. MR 1724489
[4] Bate J.A., van Rees G.H.J.: Minimal and near-minimal critical sets in back circulant latin squares. Australas. J. Combinatorics 27 (2003), 47-62. MR 1955387 | Zbl 1024.05014
[5] Cavenagh N.J.: Latin trade algorithms and the smallest critical set in a latin square. J. Autom. Lang. Combin. 8 (2003), 567-578. MR 2069074 | Zbl 1052.05019
[6] 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
[7] Cavenagh N.J.: The size of the smallest critical set in the back circulant latin square. submitted.
[8] Cavenagh N.J., Donovan D., Drápal A.: $3$-homogeneous latin trades. submitted.
[9] Conway J.C., Sloane N.J.: Sphere Packings, Lattices and Groups. New York, Springer-Verlag, 1998. Zbl 0915.52003
[10] Dénes J., Keedwell A.D.: Latin Squares and Their Applications. English Universities Press, London, 1974. MR 0351850
[11] Donovan D., Howse A., Adams P.: A discussion of latin interchanges. J. Comb. Math. Comb. Comput. 23 (1997), 161-182. MR 1432756 | Zbl 0867.05010
[12] 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
[13] Drápal A.: On a planar construction of quasigroups. Czechoslovak Math. J. 41 (1991), 538-548. MR 1117806
[14] Drápal A.: Hamming distances of groups and quasi-groups. Discrete Math. 235 (2001), 189-197. MR 1829848 | Zbl 0986.20065
[15] Drápal A.: Geometry of latin trades. manuscript circulated at the conference Loops'03, Prague 2003.
[16] Drápal, Kepka T.: Exchangeable Groupoids I. Acta Univ. Carolinae - Math. Phys. 24 (1983), 57-72. MR 0733686
[17] Drápal, Kepka T.: Exchangeable Groupoids II. Acta Univ. Carolinae - Math. Phys. 26 (1985), 3-9. MR 0830261
[18] Drápal, Kepka T.: On a distance of groups and latin squares. Comment. Math. Univ. Carolinae 30 (1989), 621-626. MR 1045889
[19] Hedayat A.S.: The theory of trade-off for $t$-designs. in ``Coding theory and design theory, Part II'', IMA Vol. Math. Appl. 21, Springer, NY, 1990. MR 1056530 | Zbl 0721.05008
[20] Horak P., Aldred R.E.L., Fleischner H.: Completing Latin squares: critical sets. J. Combin. Des. 10 (2002), 419-432. MR 1932121 | Zbl 1025.05011
[21] Keedwell A.D.: Critical sets and critical partial latin squares. in ``Proc. Third China-USA International Conf. on Graph Theory, Combinatorics, Algorithms and Applications'', World Sci. Publishing, NJ, 1994. MR 1313960
[22] 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
[23] Khodkar A.: On smallest critical sets for the elementary abelian $2$-group. Utilitas Math. 54 (1998), 45-50. MR 1658157 | Zbl 0922.05012
[24] Lütkepohl H.: Handbook of Matrices. Chichester, John Wiley and Sons, 1996. MR 1433592
[25] Street A.P.: Trades and defining sets. in: C.J. Colbourn and J.H. Dinitz, Eds., CRC Handbook of Combinatorial Designs, CRC Press, Boca Raton, FL., 1996, pp.474-478. Zbl 0847.05011
Partner of
EuDML logo