Skip to main content Accessibility help
×
Hostname: page-component-848d4c4894-xfwgj Total loading time: 0 Render date: 2024-06-15T18:37:49.387Z Has data issue: false hasContentIssue false

Transversals in Latin Squares

Published online by Cambridge University Press:  23 May 2024

Felix Fischer
Affiliation:
Queen Mary University of London
Robert Johnson
Affiliation:
Queen Mary University of London
Get access

Summary

A Latin square is an n by n grid filled with n symbols so that each symbol appears exactly once in each row and each column. A transversal in a Latin square is a collection of cells which do not share any row, column, or symbol. This survey will focus on results from the last decade which have continued the long history of the study of transversals in Latin squares.

Type
Chapter
Information
Publisher: Cambridge University Press
Print publication year: 2024

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aharoni, R. and Berger, E., Rainbow matchings in r-partite r-graphs, The Electronic Journal of Combinatorics 16 (2009), R119.CrossRefGoogle Scholar
Aharoni, R., Berger, E., Kotlar, D., and Ziv, R., On a conjecture of Stein, In: Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, vol. 87, Springer, 2017, pp. 203211.Google Scholar
Aharoni, R., Kotlar, D., and Ziv, R., Representation of large matchings in bipartite graphs, SIAM Journal on Discrete Mathematics 31 (2017), no. 3, 17261731.CrossRefGoogle Scholar
Akbari, S., Etesami, O., Mahini, H., and Mahmoody, M., On rainbow cycles in edge colored complete graphs, Australasian Journal of Combinatorics 37 (2007), 3342.Google Scholar
Alon, N. and Asodi, V., Edge colouring with delays, Combinatorics, Probability and Computing 16 (2007), no. 2, 173191.CrossRefGoogle Scholar
Alon, N., Kim, J.-H., and Spencer, J., Nearly perfect matchings in regular simple hypergraphs, Israel Journal of Mathematics 100 (1997), no. 1, 171187.CrossRefGoogle Scholar
Alon, N., Pokrovskiy, A., and Sudakov, B.. Random subgraphs of properly edgecoloured complete graphs and long rainbow cycles, Israel Journal of Mathematics 222 (2017), no. 1, 317331.CrossRefGoogle Scholar
Alon, N. and Spencer, J. H., The probabilistic method, John Wiley & Sons, 2016.Google Scholar
Andersen, L. D., Hamilton circuits with many colours in properly edge-coloured complete graphs, Mathematica Scandinavica 64 (1989), 514.CrossRefGoogle Scholar
Andersen, L. D., The history of Latin squares, Combinatorics: Ancient & Modern (edited by Wilson, R. and Watkins, J.J.), 2013.CrossRefGoogle Scholar
Balogh, J. and Molla, T., Long rainbow cycles and Hamiltonian cycles using many colors in properly edge-colored complete graphs, European Journal of Combinatorics 79 (2019), 140151.CrossRefGoogle Scholar
Barber, B., Kühn, D., Lo, A., and Osthus, D., Edge-decompositions of graphs with high minimum degree, Advances in Mathematics 288 (2016), 337385.CrossRefGoogle Scholar
Benford, A., Bowtell, C., and Montgomery, R.. Long rainbow cycles in properly coloured graphs. In preparation, 2023.Google Scholar
Best, D., Pula, K., and Wanless, I. M., Small Latin arrays have a near transversal, Journal of Combinatorial Designs 29 (2021), no. 8, 511527.CrossRefGoogle Scholar
Best, D. and Wanless, I. M., What did Ryser conjecture?, arXiv preprint arXiv:1801.02893 (2018).Google Scholar
Bohman, T., Frieze, A., and Lubetzky, E., Random triangle removal, Advances in Mathematics 280 (2015), 379438.CrossRefGoogle Scholar
Bose, R. C. and Shrikhande, S. S.. On the falsity of Euler’s conjecture about the non-existence of two orthogonal Latin squares of order 4t + 2, Proceedings of the National Academy of Sciences 45 (1959), no. 5, 734737.CrossRefGoogle ScholarPubMed
Bose, R. C., Shrikhande, S. S., and E. T. Parker, Further results on the construction of mutually orthogonal Latin squares and the falsity of Euler’s conjecture, Canadian Journal of Mathematics 12 (1960), 189203.CrossRefGoogle Scholar
Bowtell, C. and Keevash, P., The n-queens problem, arXiv preprint arXiv:2109.08083 (2021).Google Scholar
Bray, J. N., Cai, Q., Cameron, P. J., Spiga, P., and Zhang, H., The Hall–Paige conjecture, and synchronization for affine and diagonal groups, Journal of Algebra 545 (2020), 2742.CrossRefGoogle Scholar
Brègman, L. M., Some properties of nonnegative matrices and their permanents, In: Nauk, Doklady Akademii, vol. 211, Russian Academy of Sciences, 1973, pp. 2730.Google Scholar
Brouwer, A. E., On the size of a maximum transversal in a Steiner triple system, Canadian Journal of Mathematics 33 (1981), no. 5, 12021204.CrossRefGoogle Scholar
Brouwer, A. E., de Vries, A., and Wieringa, R.. A lower bound for the length of partial transversals in a Latin square, Nieuw Archief Voor Wiskunde 26 (1978), no. 2, 330332.Google Scholar
Brualdi, R. A. and Ryser, H. J., Combinatorial matrix theory, Cambridge University Press, 1991.CrossRefGoogle Scholar
Bryant, D. and Horsley, D., A second infinite family of Steiner triple systems without almost parallel classes, Journal of Combinatorial Theory, Series A, 120 (2013), no. 7, 18511854.CrossRefGoogle Scholar
Bryant, D. and Horsley, D., Steiner triple systems without parallel classes, SIAM Journal on Discrete Mathematics 29 (2015), no. 1, 693696.CrossRefGoogle Scholar
Cavenagh, N. J. and Wanless, I. M., Latin squares with no transversals, The Electronic Journal of Combinatorics (2017), P2.45.CrossRefGoogle Scholar
Colbourn, C. J. and Rosa, A., Colorings of block designs, In: Contemporary design theory: A collection of surveys, 1992, pp. 401430Google Scholar
Conlon, D., Gowers, W. T., Samotij, W., and Schacht, M., On the KLR conjecture in random graphs, Israel Journal of Mathematics 203 (2014), no. 1, 535580.CrossRefGoogle Scholar
Drake, D. A., Maximal sets of Latin squares and partial transversals, Journal of Statistical Planning and Inference 1 (1977), no. 2, 143149.CrossRefGoogle Scholar
Eberhard, S., Manners, F., and Mrazović, R., An asymptotic for the Hall–Paige conjecture, Advances in Mathematics 404 (2022), 108423.CrossRefGoogle Scholar
Eberhard, S., Manners, F., and Mrazović, R., Transversals in quasirandom Latin squares, Proceedings of the London Mathematical Society 127 (2023), no. 1, 84115.CrossRefGoogle Scholar
Egorychev, G. P., The solution of van der Waerden’s problem for permanents, Advances in Mathematics 42 (1981), no. 3, 299305.CrossRefGoogle Scholar
Ehard, S., Glock, S., and Joos, F., Pseudorandom hypergraph matchings, Combinatorics, Probability and Computing 29 (2020), no. 6, 868885.CrossRefGoogle Scholar
Erdős, P., Gyárfás, A., and Pyber, L., Vertex coverings by monochromatic cycles and trees, Journal of Combinatorial Theory, Series B 51 (1991), no. 1, 9095.CrossRefGoogle Scholar
Euler, L., Recherches sur un nouvelle espéce de quarrés magiques, Verhandelingen uitgegeven door het zeeuwsch Genootschap der Wetenschappen te Vlissingen 9 (1782), 85239.Google Scholar
Evans, A. B., Latin squares without orthogonal mates, Designs, Codes and Cryptography 40 (2006), 121130.CrossRefGoogle Scholar
Evans, A. B., The admissibility of sporadic simple groups, Journal of Algebra 321 (2009), no. 1, 105116.CrossRefGoogle Scholar
. Orthogonal Latin squares based on groups, Springer, 2018.Google Scholar
Falikman, D. I., Proof of the van der Waerden conjecture regarding the permanent of a doubly stochastic matrix, Mathematical Notes of the Academy of Sciences of the USSR 29 (1981), 475479.Google Scholar
Ferber, A., Kronenberg, G., and Long, E., Packing, counting and covering Hamilton cycles in random directed graphs, Israel Journal of Mathematics 220 (2017), 5787.CrossRefGoogle Scholar
Ferber, A. and Kwan, M., Almost all Steiner triple systems are almost resolvable, Forum of Mathematics, Sigma 8 (2020), 39.CrossRefGoogle Scholar
Friedlander, R. J., Gordon, B., and M. D. Miller, On a group sequencing problem of Ringel, Congr. Numer 21 (1978), 307321.Google Scholar
Georgakopoulos, A., Delay colourings of cubic graphs, The Electronic Journal of Combinatorics (2013), P45.CrossRefGoogle Scholar
Glebov, R. and Luria, Z., On the maximum number of Latin transversals, Journal of Combinatorial Theory, Series A 141 (2016), 136146.CrossRefGoogle Scholar
Glock, S., Kühn, D., Montgomery, R., and Osthus, D., Decompositions into isomorphic rainbow spanning trees, Journal of Combinatorial Theory, Series B 146 (2021), 439484.CrossRefGoogle Scholar
Gould, S. and Kelly, T., Hamilton transversals in random Latin squares, Random Structures & Algorithms 62 (2023), no. 2, 450478.CrossRefGoogle Scholar
Gould, S., Kelly, T., Kühn, D., and Osthus, D., Almost all optimally coloured complete graphs contain a rainbow Hamilton path, Journal of Combinatorial Theory, Series B 156 (2022), 57100.CrossRefGoogle Scholar
Gowers, W. T., Quasirandom groups, Combinatorics, Probability and Computing 17 (2008), no. 3, 363387.CrossRefGoogle Scholar
Hahn, G., Un jeu de colouration, In: Actes du Colloque de Cerisy, vol. 12, 1980, pp. 1818.Google Scholar
Hall, M. and Paige, L. J., Complete mappings of finite groups, Pacific Journal of Mathematics 5 (1955), no. 4, 541549.CrossRefGoogle Scholar
Hall, M. Jr., Combinatorial Theory, Waltham, Mass., USA, 1967.Google Scholar
Hatami, P. and Shor, P. W., A lower bound for the length of a partial transversal in a Latin square, Journal of Combinatorial Theory, Series A 115 (2008), no. 7, 11031113.CrossRefGoogle Scholar
Janson, S. and A. Ruciński, The infamous upper tail, Random Structures & Algorithms 20 (2002), no. 3, 317342.CrossRefGoogle Scholar
Kang, D. Y., Kelly, T., Kühn, D., Methuku, A., and Osthus, D., Graph and hypergraph colouring via nibble methods: A survey, arXiv preprint arXiv:2106.13733 (2021).Google Scholar
Keevash, P., The existence of designs, arXiv preprint arXiv:1401.3665 (2014).Google Scholar
Keevash, P., Counting designs, Journal of the European Mathematical Society 20 (2018), no. 4, 903927.CrossRefGoogle Scholar
Keevash, P., Pokrovskiy, A., Sudakov, B., and Yepremyan, L., New bounds for Ryser’s conjecture and related problems, Transactions of the American Mathematical Society, Series B, 9 (2022), no. 8, 288321.CrossRefGoogle Scholar
Kirkman, T., On a problem in combinatorics, Cambridge and Dublin Math J. 2 (1847), no. 1847, 191204.Google Scholar
Kohayakawa, Y., Nagle, B., Rödl, V., and Schacht, M., Weak hypergraph regularity and linear hypergraphs, Journal of Combinatorial Theory, Series B 100 (2010), no. 2, 151160.CrossRefGoogle Scholar
Kohayakawa, Y. and V. Rödl, Szemerédi’s regularity lemma and quasirandomness, In: Recent advances in algorithms and combinatorics, 2003, pp. 289351.CrossRefGoogle Scholar
Koksma, K. K., A lower bound for the order of a partial transversal in a Latin square, Journal of Combinatorial Theory 7 (1969), no. 1, 9495.CrossRefGoogle Scholar
Komlós, J. and Szemerédi, E., Topological cliques in graphs, Combinatorics, Probability and Computing 3 (1994), no. 2, 247256.CrossRefGoogle Scholar
Komlós, J. and Szemerédi, E., Topological cliques in graphs II, Combinatorics, Probability and Computing 5 (1996), no. 1, 7990.CrossRefGoogle Scholar
Krivelevich, M., Triangle factors in random graphs, Combinatorics, Probability and Computing 6 (1997), no. 3, 337347.CrossRefGoogle Scholar
Kwan, M., Almost all Steiner triple systems have perfect matchings, Proceedings of the London Mathematical Society 121 (2020), no. 6, 14681495.CrossRefGoogle Scholar
Kwan, M., Sah, A., and Sawhney, M., Large deviations in random Latin squares, Bulletin of the London Mathematical Society 54 (2022), no. 4, 14201438.CrossRefGoogle Scholar
Kwan, M., Sah, A., Sawhney, M., and Simkin, M., Substructures in Latin squares, arXiv preprint arXiv:2202.05088 (2022).Google Scholar
Kwan, M. and Sudakov, B., Intercalates and discrepancy in random Latin squares, Random Structures & Algorithms 52 (2018), no. 2, 181196.CrossRefGoogle Scholar
Letzter, S., Sublinear expanders and their applications, In: Surveys in Combinatorics 2024, Cambridge University Press, 2024.Google Scholar
Lindner, C. C. and Phelps, K. T., A note on partial parallel classes in Steiner systems, Discrete Mathematics 24 (1978), no. 1, 109112.CrossRefGoogle Scholar
Maamoun, M. and Meyniel, H., On a problem of G. Hahn about coloured Hamiltonian paths in K2t, Discrete Mathematics 51 (1984), no. 2, 213214.CrossRefGoogle Scholar
Maillet, E., Sur les carrés latins déuler, Assoc. Franc. Caen 23 (1894), 244252.Google Scholar
Mann, H. B., On orthogonal Latin squares, Bulletin of the American Mathematical Society 50 (1944), no. 4, 249257.CrossRefGoogle Scholar
McKay and I. M. Wanless, B. D., Most Latin squares have many subsquares, Journal of Combinatorial Theory, Series A 86 (1999), no. 2, 323347.CrossRefGoogle Scholar
Montgomery, R., A proof of the Ryser-Brualdi-Stein conjecture for large even n, In preparation.Google Scholar
Montgomery, R., Spanning trees in random graphs, Advances in Mathematics 356 (2019), 106793.CrossRefGoogle Scholar
Montgomery, R., Pokrovskiy, A., and Sudakov, B., Decompositions into spanning rainbow structures, Proceedings of the London Mathematical Society 119 (2019), no. 4, 899959.CrossRefGoogle Scholar
Montgomery, R., Pokrovskiy, A., and Sudakov, B., A proof of Ringel’s conjecture, Geometric and Functional Analysis 31 (2021), no. 3, 663720.CrossRefGoogle Scholar
Morris, P.. Random Steiner triple systems. Master’s thesis, Freie Universität Berlin, 2017.Google Scholar
Munhá Correia, D., Pokrovskiy, A., and Sudakov, B., Short proofs of rainbow matchings results, International Mathematics Research Notices 2023 (2023), no. 14, 1244112476.CrossRefGoogle Scholar
Müyesser, A., Cycle type in Hall-Paige: A proof of the Friedlander-Gordon-Tannenbaum conjecture, arXiv preprint arXiv:2303.16157 (2023).Google Scholar
Müyesser, A. and Pokrovskiy, A., A random Hall-Paige conjecture, arXiv preprint arXiv:2204.09666 (2022).Google Scholar
Ozanam, J., Récreátions mathématiques et physiques, vol. 1, 1723.Google Scholar
Paige, L. J., A note on finite abelian groups, Bulletin of the American Mathematical Society 53 (1947), no. 6, 590593.CrossRefGoogle Scholar
Paige, L. J., Complete mappings of finite groups, Pacific J. Math. 1 (1951), no. 1, 111116.CrossRefGoogle Scholar
Parker, E. T., Pathological Latin squares, Proc. Symbs. Pure Math 19 (1971), 177181.CrossRefGoogle Scholar
Pippenger, N. and Spencer, J., Asymptotic behavior of the chromatic index for hypergraphs, Journal of Combinatorial Theory, Series A 51 (1989), no. 1, 2442.CrossRefGoogle Scholar
Pokrovskiy, A., An approximate version of a conjecture of Aharoni and Berger, Advances in Mathematics 333 (2018), 11971241.CrossRefGoogle Scholar
Pokrovskiy, A., Rainbow Subgraphs and their Applications, In: Surveys in Combinatorics 2022, Cambridge University Press, 2022, pp. 191214.Google Scholar
Pokrovskiy, A. and Sudakov, B., A counterexample to Stein’s equi-n-square conjecture, Proceedings of the American Mathematical Society 147 (2019), no. 6, 22812287.CrossRefGoogle Scholar
Ray-Chaudhuri, D. K. and R. M. Wilson, Solution of Kirkman’s schoolgirl problem, In: Proceedings of Symposia in Pure Mathematics, vol. 19, 1971, pp. 187203.CrossRefGoogle Scholar
Rödl, V., On a packing and covering problem, European Journal of Combinatorics 6 (1985), no. 1, 6978.CrossRefGoogle Scholar
Rödl, V. and A. Ruciński, Threshold functions for Ramsey properties, Journal of the American Mathematical Society 8 (1995), no. 4, 917942.CrossRefGoogle Scholar
Rödl, V., Ruciński, A., and E. Szemerédi, A Dirac-type theorem for 3-uniform hypergraphs, Combinatorics, Probability and Computing 15 (2006), no. 1-2, 229251.CrossRefGoogle Scholar
Ryser, H., Neuere Probleme der Kombinatorik, In: Vorträge über Kombinatorik, Oberwolfach, 1967, pp. 6991.Google Scholar
Shor, P. W., A lower bound for the length of a partial transversal in a Latin square, Journal of Combinatorial Theory, Series A 33 (1982), no. 1, 18.CrossRefGoogle Scholar
Stein, S. K., Transversals of Latin squares and their generalizations, Pacific J. Math. 59 (1975), 567575.CrossRefGoogle Scholar
Taranenko, A., Multidimensional permanents and an upper bound on the number of transversals in Latin squares, Journal of Combinatorial Designs 23 (2015), no. 7, 305320.CrossRefGoogle Scholar
Tarry, G., Le problème des 36 officiers, Secrétariat de lássociation française pour lávancement des sciences, 1900.Google Scholar
Van Lint, J. H. and Wilson, R. M., A course in combinatorics, Cambridge University Press, 2001.CrossRefGoogle Scholar
van Rees, G., Subsquares and transversals in Latin squares, Ars Combinatoria 29 (1990), 193204.Google Scholar
Wang, S. P., On self-orthogonal Latin squares and partial transversals of Latin squares, PhD thesis, The Ohio State University, 1978.Google Scholar
Wanless, I. M., Transversals in Latin squares: A survey, In: Surveys in Combinatorics 2011, Cambridge University Press, 2011, pp. 403437.CrossRefGoogle Scholar
Wanless, I. M. and Webb, B. S., The existence of Latin squares without orthogonal mates, Designs, Codes and Cryptography 40 (2006), 131135.CrossRefGoogle Scholar
Wilcox, S., Reduction of the Hall–Paige conjecture to sporadic simple groups, Journal of Algebra 321 (2009), no. 5, 14071428.CrossRefGoogle Scholar
Woolbright, D. E., An n × n Latin square has a transversal with at least n − √n distinct symbols, Journal of Combinatorial Theory, Series A 24 (1978), no. 2, 235237.CrossRefGoogle Scholar

Save book to Kindle

To save this book to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×