| Title:
|
On elementary moves that generate all spherical latin trades (English) |
| Author:
|
Drápal, Aleš |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
50 |
| Issue:
|
4 |
| Year:
|
2009 |
| Pages:
|
477-511 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We show how to generate all spherical latin trades by elementary moves from a base set. If the base set consists only of a single trade of size four and the moves are applied only to one of the mates, then three elementary moves are needed. If the base set consists of all bicyclic trades (indecomposable latin trades with only two rows) and the moves are applied to both mates, then one move suffices. Many statements of the paper pertain to all latin trades, not only to spherical ones. (English) |
| Keyword:
|
latin trade |
| Keyword:
|
spherical latin bi-trade |
| Keyword:
|
planar Eulerian triangulation |
| MSC:
|
05B15 |
| idZBL:
|
Zbl 1200.05036 |
| idMR:
|
MR2583128 |
| . |
| Date available:
|
2009-12-22T10:02:49Z |
| Last updated:
|
2013-09-22 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/137441 |
| . |
| Reference:
|
[1] Batagelj V.: An improved inductive definition of two restricted classes of triangulations of the plane.Combinatorics and graph theory (Warsaw 1987), 11--18, Banach Center Publ., 25, PWN, Warsaw, 1989. Zbl 0742.05033, MR 1097631 |
| Reference:
|
[2] Cavenagh N., Donovan D., Drápal A.: $3$-homogeneous latin trades.Discrete Math. 300 (2005), 57--70. MR 2170114, 10.1016/j.disc.2005.04.021 |
| Reference:
|
[3] Cavenagh N.J., Donovan D., Drápal A.: $4$-homogeneous latin trades.Australas. J. Combin. 32 (2005), 285--303. MR 2139816 |
| Reference:
|
[4] Cavenagh N.J., Hämäläinen C., Drápal A.: Latin bitrades derived from groups.Discrete Math. 308 (2008), 6189--6202. MR 2464907, 10.1016/j.disc.2007.11.041 |
| Reference:
|
[5] Cavenagh N.J., Lisoněk P.: Planar Eulerian triangulations are equivalent to spherical Latin bitrades.J. Combin. Theory Ser. A 115 (2008), 193--197. MR 2378864, 10.1016/j.jcta.2007.04.002 |
| Reference:
|
[6] Cavenagh N.J., Wanless I.M.: Latin trades in groups defined on planar triangulations.J. Algebr. Comb. (in print), DOI 10.1007/s10801-008-0165-9. |
| Reference:
|
[7] Drápal A., Kepka T.: Exchangeable partial groupoids I.Acta Univ. Carolin. Math. Phys. 24 (1983), 57--72. MR 0733686 |
| Reference:
|
[8] Drápal A., Kepka T.: Group modifications of some partial groupoids.Ann. Discrete Math. 18 (1983), 319--332. MR 0695819 |
| Reference:
|
[9] Drápal A.: On a planar construction of quasigroups.Czechoslovak Math. J. 41 (1991), 538--548. MR 1117806 |
| Reference:
|
[10] Drápal A.: Latin Squares and Partial Groupoids.(in Czech), Candidate of Science Thesis, Charles University, Prague, 1988. |
| Reference:
|
[11] Drápal A.: Hamming distances of groups and quasi-groups.Discrete Math. 235 (2001), 189--197. MR 1829848, 10.1016/S0012-365X(00)00272-7 |
| Reference:
|
[12] Drápal A.: Geometry of Latin Trades.manuscript circulated at the conference Loops'03, Prague, 2003. |
| Reference:
|
[13] Drápal A.: Geometrical structure and construction of latin trades.Adv. Geom. 9 (2009), 311--348. MR 2537024, 10.1515/ADVGEOM.2009.018 |
| Reference:
|
[14] Drápal A., Hämäläinen C., Kala V.: Latin bitrades, dissections of equilateral triangles and abelian groups.J. Comb. Des. (in print), DOI 10.1002/jcd.20237. |
| Reference:
|
[15] Drápal A., Lisoněk P.: Generating spherical Eulerian triangulations.Discrete Math.(to appear). MR 2592497 |
| Reference:
|
[16] Grannell M.J., Griggs T.S., Knor M.: Biembeddings of symmetric configurations and $3$-homogeneous Latin trades.Comment. Math. Univ. Carolin. 49 (2008), 411--420. MR 2490436 |
| Reference:
|
[17] Hämäläinen C.: Partitioning $3$-homogeneous latin bitrades.Geom. Dedicata 133 (2008), 181--193. MR 2390076, 10.1007/s10711-008-9242-4 |
| Reference:
|
[18] Heawood P.J.: On the four colour map theorem.Quart. J. 29 (1898), 270--285. |
| Reference:
|
[19] Holton D.A., Manvel B., McKay B.D.: Hamiltonian cycles in cubic $3$-connected bipartite planar graphs.J. Combin. Theory Ser. B 38 (1985), 279--297. Zbl 0551.05052, MR 0796604, 10.1016/0095-8956(85)90072-3 |
| Reference:
|
[20] 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:
|
[21] Lefevre J., Cavenagh N.J., Donovan D., Drápal A.: Minimal and minimum size latin bitrades of each genus.Comment. Math. Univ. Carolin. 48 (2007), 189--203. MR 2338087 |
| Reference:
|
[22] Lefevre J.G., Donovan D., Drápal A.: Permutation representation of $3$ and $4$-homogenous latin bitrades.Fund. Inform. 84 (2008), 99--110. MR 2422431 |
| . |
