[1] Abel R. J. R., Li Y.:
Some constructions for $t$ pairwise orthogonal diagonal Latin squares based on difference matrices. Discrete Math. 338 (2015), no. 4, 593–607.
DOI 10.1016/j.disc.2014.11.018 |
MR 3300747
[2] Andersen L. D., Hilton A. J. W.:
Thanks Evans!. Proc. London Math. Soc. (3) 47 (1983), no. 3, 507–522.
MR 0716801
[3] Andersen L. D., Hilton A. J. W., Rodger C. A.:
A solution to the embedding problem for partial idempotent Latin squares. J. London Math. Soc. (2) 26 (1982), no. 1, 21–27.
DOI 10.1112/jlms/s2-26.1.21 |
MR 0667240
[4] Barber B., Kühn D., Lo A., Osthus D., Taylor A.:
Clique decompositions of multipartite graphs and completion of Latin squares. J. Combin. Theory Ser. A. 151 (2017), 146–201.
DOI 10.1016/j.jcta.2017.04.005 |
MR 3663493
[5] Belyavskaya G. B., Lumpov A. D.:
Cross product of two systems of quasigroups and its use in constructing partially orthogonal quasigroups. Mat. Issled., Issled. Teor. Binarnykh i $n$-arnykh Kvazigrupp 83 (1985), 26–38 (Russian).
MR 0807271
[7] Bryant D., Horsley D.:
A proof of Lindner's conjecture on embeddings of partial Steiner triple systems. J. Comb. Des. 17 (2009), no. 1, 63–89.
DOI 10.1002/jcd.20189 |
MR 2475426
[9] Colbourn C. J., Dinitz J. H.:
Handbook of Combinatorial Designs. Chapman and Hall/CRC, 2007.
MR 2246267
[10] Colbourn C. J., Zhu L.:
The spectrum of $R$-orthogonal Latin squares. Combinatorics Advances, Tehran, 1994, Math. Appl., 329, Kluwer Acad. Publ., Dordrecht, 1995, pages 49–75.
MR 1366841
[12] Damerell R. M.:
On Smetaniuk's construction for Latin squares and the Andersen–Hilton theorem. Proc. London Math. Soc. (3) 47 (1983), no. 3, 523–526.
MR 0716802
[13] Dietrich H., Wanless I. M.:
Small partial Latin squares that embed in an infinite group but not into any finite group. J. Symbolic Comput. 86 (2018), 142–152.
DOI 10.1016/j.jsc.2017.04.002 |
MR 3725217
[14] Donovan D., Grannell M., Yazıcı E. Ş.:
Embedding partial Latin squares in Latin squares with many mutually orthogonal mates. Discrete Math. 343 (2020), no. 6, 111835, 6 pages.
DOI 10.1016/j.disc.2020.111835 |
MR 4062293
[16] Drake D. A., Lenz H.:
Orthogonal Latin squares with orthogonal subsquares. Arch. Math. (Basel) 34 (1980), no. 6, 565–576.
DOI 10.1007/BF01225000 |
MR 0596867
[17] Evans A. B.:
Orthomorphism Graphs of Groups. Lecture Notes in Mathematics, 1535, Springer, Berlin, 1992.
MR 1222645
[19] Falcón R. M., Falcón Ó. J., Núñez J.: Computing the sets of totally symmetric and totally conjugate orthogonal partial Latin squares by means of a SAT solver. Proc. of 17th Int. Conf. Computational and Mathematical Methods in Science and Engineering, CMMSE 2017, pages 841–852.
[20] Ganter B.:
Endliche Vervollständigung endlicher partieller Steinerscher Systeme. Arch. Math. (Basel) 22 (1971), 328–332 (German).
DOI 10.1007/BF01222584 |
MR 0294145
[21] Ganter B.:
Partial pairwise balanced designs. Colloq. Int. Sulle Teorie Combinatorie, Rome, 1973, Tomo II, Accad. Naz. Lincei, 1976, pages 377–380.
MR 0472553
[22] Gustavsson T.: Decompositions of Large Graphs and Digraphs with High Minimum Degree. Ph.D. Thesis, Stockholm University, Stockholm, 1991.
[24] Hall P.: On representative subsets. Classic Papers in Combinatorics, Birkhäuser, Boston, 1987, pages 58–62.
[26] Hilton A. J. W., Rodger C. A., Wojciechowski J.:
Prospects for good embeddings of pairs of partial orthogonal Latin squares and of partial Kirkman triple systems. J. Combin. Math. Combin. Comput. 11 (1992), 83–91.
MR 1160067
[27] Hirsch R., Jackson M.:
Undecidability of representability as binary relations. J. Symbolic Logic 77 (2012), no. 4, 1211–1244.
DOI 10.2178/jsl.7704090 |
MR 3051622
[29] Jenkins P.:
Embedding a restricted class of partial $K_4$ designs. Ars Combin. 77 (2005), 295–303.
MR 2180852
[30] Jenkins P.:
Embedding a Latin square in a pair of orthogonal Latin squares. J. Combin. Des. 14 (2006), no. 4, 270–276.
DOI 10.1002/jcd.20087 |
MR 2229878
[31] Jenkins P.:
Partial graph design embeddings and related problems. Bull. Austral. Math. Soc. 73 (2006), 159–160.
DOI 10.1017/S0004972700038715
[32] Keevash P.:
Coloured and directed designs. I. Bárány, G. Katona, A. Sali eds., Building Bridges II., Bolyai Society Mathematical Studies, 28, Springer, Berlin, 2019.
DOI 10.1007/978-3-662-59204-5_9
[33] König D.:
Über Graphen und ihre Anwendungen auf Determinantentheorie und Mengenlehre. Math. Ann. 77 (1916), no. 4, 453–465 (German).
DOI 10.1007/BF01456961 |
MR 1511872
[34] Luria Z.: New bounds on the number of $n$-queens configurations. available at arXiv: 1705.05225v2 [math.CO] (2017), 12 pages.
[37] Lindner C. C.:
A survey of finite embedding theorems for partial Latin squares and quasigroups. Graphs and Combinatorics, Lecture Notes in Math., 406, Springer, Berlin, 1974, pages 109–152.
DOI 10.1007/BFb0066437 |
MR 0379211
[38] Lindner C. C.:
A partial Steiner triple system of order $n$ can be embedded in a Steiner triple system of order $6n+3$. J. Comb. Theory Ser. A. 18 (1975), 349–351.
DOI 10.1016/0097-3165(75)90046-1 |
MR 0379223
[40] Lindner C. C., Cruse A. B.:
Small embeddings for partial semisymmetric and totally symmetric quasigroups. J. London Math. Soc. (2) 12 (1976), 479–484.
DOI 10.1112/jlms/s2-12.4.479 |
MR 0432796
[43] Nosov V. A., Sachkov V. N., Tarakanov V. E.:
Combinatorial analysis (matrix problems, the theory of sampling). Probability Theory. Mathematical Statistics. Theoretical Cybernetics. 188, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Informatsii, Moscow 18 (1981), 53–93, 188 (Russian).
MR 0625132
[44] Quackenbush R. W.:
Near vector spaces over GF$(q)$ and $(v,q+1,1)$-BIBDs. Linear Algebra Appl. 10 (1975), 259–266.
MR 0369099
[46] Rodger C. A.:
Recent results on the embedding of Latin squares and related structures, cycle systems and graph designs. Matematiche (Catania) 47 (1992), no. 2, 295–311.
MR 1275861
[48] Smetaniuk B.:
A new construction on Latin squares. I. A proof of the Evans conjecture. Ars Combin. 11 (1981), 155–172.
MR 0629869
[50] Treash A. C.:
Inverse Property Loops and Related Steiner Triple Systems. Ph.D. Thesis, Emory University, Atlanta, 1969.
MR 2618359
[52] Vodička M., Zlatoš P.:
The finite embeddability property for IP loops and local embeddability of groups into finite IP loops. Ars Math. Contemp. 17 (2019), no. 2, 535–554.
DOI 10.26493/1855-3974.1884.5cb |
MR 4041359
[53] Van der Waerden B. L.:
Ein Satz über Klasseneinteilungen von endlichen Mengen. Abh. Math. Sem. Univ. Hamburg 5 (1927), no. 1, 185–188 (German).
DOI 10.1007/BF02952519 |
MR 3069474
[54] Wallis W. D., Zhu L.:
Orthogonal Latin squares with small subsquares. Combinatorial Mathematics, X, Adelaide, 1982, Lecture Notes in Math., 1036, Springer, Berlin, 1983, pages 398–409.
MR 0731596
[55] Wanless I. M., Webb B. S.:
Small partial Latin squares that cannot be embedded in a Cayley table. Australas. J. Combin. 67 (2017), no. 2, 352–363.
MR 3607832
[57] Zhu L.:
Some results on orthogonal Latin squares with orthogonal subsquares. Utilitas Math. 25 (1984), 241–248.
MR 0752862