Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-28T15:27:08.164Z Has data issue: false hasContentIssue false
  • English
  • Français

Orthogonal Latin Rectangles

Published online by Cambridge University Press:  01 July 2008

ROLAND HÄGGKVIST  and
ANDERS JOHANSSON
ROLAND HÄGGKVIST
Affiliation:
Matematiska Institutionen, Umeå Universitet, S-901 87 Umeå, Sweden (e-mail: [email protected])
ANDERS JOHANSSON
Affiliation:
N-Institutionen, Högskolan i Gävle, S-801 76 Gävle, Sweden (e-mail: [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We use a greedy probabilistic method to prove that, for every ε > 0, every m × n Latin rectangle on n symbols has an orthogonal mate, where m = (1 − ε)n. That is, we show the existence of a second Latin rectangle such that no pair of the mn cells receives the same pair of symbols in the two rectangles.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 17 , Issue 4 , July 2008 , pp. 519 - 536
Copyright
Copyright © Cambridge University Press 2008

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

[1]Alon, N. and Spencer, J. (1992) The Probabilistic Method, Wiley.Google Scholar
[2]Birkhoff, G. (1946) Three observations on linear algebra. Univ. Nac. Tucumán. Rev. Ser. A 5 147–151.Google Scholar
[3]Brouwer, A. E., de Vries, A. J. and Wieringa, R. M. A (1978) A lower bound for the length of a partial transversal in a Latin square. Nieuw Archief vor Wieskunde (3) XXVI 330–332.Google Scholar
[4]Dénes, J. and Keedwell, A. D., eds (1974) Latin Squares and their Applications, Academic Press, New York.Google Scholar
[5]Drisko, A. A. (1998) Transversals in row-Latin rectangles. J. Combin. Theory Ser. A 84 181–195.CrossRefGoogle Scholar
[6]Hilton, A. J. W. (1994) Problem BCC 13.20. Discrete Math. 125 407–417.Google Scholar
[7]Jensen, T. R. and Toft, B. (1995) Graph Colouring Problems, Wiley, New York.Google Scholar
[8]Kahn, J. (1996) Asymptotically good list-colourings. J. Combin. Theory. Ser. A 73 1–59.CrossRefGoogle Scholar
[9]Koksma, K. K. (1969) A lower bound for the order of a partial transversal in a Latin square. J. Combin. Theory 7 94–95.CrossRefGoogle Scholar
[10]Molloy, M. S. and Reed, B. (2002) Graph Colouring and the Probabilistic Method, Springer.CrossRefGoogle Scholar
[11]Shor, P. W. (1982) A lower bound for the length of a partial transversal in a Latin square. J. Combin. Theory Ser. A 33 1–8.CrossRefGoogle Scholar
[12]Stein, S. K. (1975) Transversals of Latin squares and their generalizations. Pacific J. Math. 59 567–575.CrossRefGoogle Scholar
[13]Woolbright, D. E. (1978) A n × n Latin square has a transversal with at least distinct symbols. J. Combin. Theory Ser. A 24 235–237.CrossRefGoogle Scholar