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Keywords:
topological embedding; torus; Klein bottle; 6-regular graph; symmetric configuration of triples; partial Latin square; 3-homogeneous Latin trade
Summary:
Using results of Altshuler and Negami, we present a classification of biembeddings of symmetric configurations of triples in the torus or Klein bottle. We also give an alternative proof of the structure of 3-homogeneous Latin trades.
References:
[1] Altshuler A.: Construction and enumeration of regular maps on the torus. Discrete Math. 115 (1973), 201-217. DOI 10.1016/S0012-365X(73)80002-0 | MR 0321797 | Zbl 0253.05117
[2] Cavenagh N.J.: A uniqueness result for $3$-homogeneous Latin trades. Comment. Math. Univ. Carolin. 47 (2006), 337-358. MR 2241536 | Zbl 1138.05007
[3] Cavenagh N.J., Donovan D.M., Drápal A.: $3$-homogeneous Latin trades. Discrete Math. 300 (2005), 57-70. DOI 10.1016/j.disc.2005.04.021 | MR 2170114 | Zbl 1073.05012
[4] Colbourn C.J., Rosa A.: Triple Systems. Clarendon Press, New York, 1999, ISBN: 0-19-853576-7. MR 1843379 | Zbl 1030.05017
[5] Donovan D.M., Drápal A., Lefevre J.G.: Permutation representation of $3$ and $4$-homogeneous Latin bitrades. submitted.
[6] Figueroa-Centeno R.M., White A.T.: Topological models for classical configurations. J. Statist. Plann. Inference 86 (2000), 421-434. DOI 10.1016/S0378-3758(99)00122-6 | MR 1768283 | Zbl 0973.05014
[7] Grannell M.J., Griggs T.S.: Designs and topology. in Surveys in Combinatorics 2007, London Math. Soc. Lecture Note Series 346, Cambridge University Press, Cambridge, 2007, pp.121-174. MR 2252792
[8] Grannell M.J., Griggs T.S., Knor M.: Biembeddings of Latin squares and Hamiltonian decompositions. Glasgow Math. J. 46 (2004), 443-457. DOI 10.1017/S0017089504001922 | MR 2094802 | Zbl 1062.05030
[9] Grannell M.J., Griggs T.S., Knor M.: Biembeddings of symmetric configurations of triples. Proceedings of MaGiA conference, Kočovce 2004, Slovak University of Technology, 2004, pp.106-112.
[10] Hämäläinen C.: Partitioning $3$-homogeneous latin bitrades. preprint. MR 2390076
[11] Kirkman T.P.: On a problem of combinations. Cambridge and Dublin Math. J. 2 (1847), 191-204.
[12] Lawrencenko S., Negami S.: Constructing the graphs that triangulate both the torus and the Klein bottle. J. Combin. Theory Ser. B 77 (1999), 211-218. DOI 10.1006/jctb.1999.1920 | MR 1710539 | Zbl 1025.05018
[13] Lefevre J.G., Donovan D.M., Grannell M.J., Griggs T.S.: A constraint on the biembedding of Latin squares. submitted. Zbl 1170.05017
[14] Negami S.: Uniqueness and faithfulness of embedding of toroidal graphs. Discrete Math. 44 (1983), 161-180. DOI 10.1016/0012-365X(83)90057-2 | MR 0689809 | Zbl 0508.05033
[15] Negami S.: Classification of $6$-regular Klein-bottlal graphs. Research Reports on Information Sciences, Department of Information Sciences, Tokyo Institute of Technology A-96 (1984), 16pp.
[16] White A.T.: Modelling finite geometries on surfaces. Discrete Math. 244 (2002), 479-493. DOI 10.1016/S0012-365X(01)00069-3 | MR 1844056 | Zbl 0989.05025
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