On elementary moves that generate all spherical latin trades

Aleš Drápal

Commentationes Mathematicae Universitatis Carolinae (2009)

  • Volume: 50, Issue: 4, page 477-511
  • ISSN: 0010-2628

Abstract

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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.

How to cite

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Drápal, Aleš. "On elementary moves that generate all spherical latin trades." Commentationes Mathematicae Universitatis Carolinae 50.4 (2009): 477-511. <http://eudml.org/doc/35125>.

@article{Drápal2009,
abstract = {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.},
author = {Drápal, Aleš},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {latin trade; spherical latin bi-trade; planar Eulerian triangulation; Latin trade; spherical Latin bi-trade; planar Eulerian triangulation},
language = {eng},
number = {4},
pages = {477-511},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On elementary moves that generate all spherical latin trades},
url = {http://eudml.org/doc/35125},
volume = {50},
year = {2009},
}

TY - JOUR
AU - Drápal, Aleš
TI - On elementary moves that generate all spherical latin trades
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2009
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 50
IS - 4
SP - 477
EP - 511
AB - 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.
LA - eng
KW - latin trade; spherical latin bi-trade; planar Eulerian triangulation; Latin trade; spherical Latin bi-trade; planar Eulerian triangulation
UR - http://eudml.org/doc/35125
ER -

References

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  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. Zbl0742.05033MR1097631
  2. Cavenagh N., Donovan D., Drápal A., 10.1016/j.disc.2005.04.021, Discrete Math. 300 (2005), 57--70. MR2170114DOI10.1016/j.disc.2005.04.021
  3. Cavenagh N.J., Donovan D., Drápal A., 4 -homogeneous latin trades, Australas. J. Combin. 32 (2005), 285--303. MR2139816
  4. Cavenagh N.J., Hämäläinen C., Drápal A., 10.1016/j.disc.2007.11.041, Discrete Math. 308 (2008), 6189--6202. MR2464907DOI10.1016/j.disc.2007.11.041
  5. Cavenagh N.J., Lisoněk P., 10.1016/j.jcta.2007.04.002, J. Combin. Theory Ser. A 115 (2008), 193--197. MR2378864DOI10.1016/j.jcta.2007.04.002
  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. 
  7. Drápal A., Kepka T., Exchangeable partial groupoids I, Acta Univ. Carolin. Math. Phys. 24 (1983), 57--72. MR0733686
  8. Drápal A., Kepka T., Group modifications of some partial groupoids, Ann. Discrete Math. 18 (1983), 319--332. MR0695819
  9. Drápal A., On a planar construction of quasigroups, Czechoslovak Math. J. 41 (1991), 538--548. MR1117806
  10. Drápal A., Latin Squares and Partial Groupoids, (in Czech), Candidate of Science Thesis, Charles University, Prague, 1988. 
  11. Drápal A., 10.1016/S0012-365X(00)00272-7, Discrete Math. 235 (2001), 189--197. MR1829848DOI10.1016/S0012-365X(00)00272-7
  12. Drápal A., Geometry of Latin Trades, manuscript circulated at the conference Loops'03, Prague, 2003. 
  13. Drápal A., 10.1515/ADVGEOM.2009.018, Adv. Geom. 9 (2009), 311--348. MR2537024DOI10.1515/ADVGEOM.2009.018
  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. 
  15. Drápal A., Lisoněk P., Generating spherical Eulerian triangulations, Discrete Math.(to appear). MR2592497
  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. MR2490436
  17. Hämäläinen C., 10.1007/s10711-008-9242-4, Geom. Dedicata 133 (2008), 181--193. MR2390076DOI10.1007/s10711-008-9242-4
  18. Heawood P.J., On the four colour map theorem, Quart. J. 29 (1898), 270--285. 
  19. Holton D.A., Manvel B., McKay B.D., 10.1016/0095-8956(85)90072-3, J. Combin. Theory Ser. B 38 (1985), 279--297. Zbl0551.05052MR0796604DOI10.1016/0095-8956(85)90072-3
  20. Keedwell A.D., Critical sets in latin squares and related matters: an update, Util. Math. 65 (2004), 97--131. Zbl1053.05019MR2048415
  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. MR2338087
  22. Lefevre J.G., Donovan D., Drápal A., Permutation representation of 3 and 4 -homogenous latin bitrades, Fund. Inform. 84 (2008), 99--110. MR2422431

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