Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-19T15:49:17.197Z Has data issue: false hasContentIssue false
  • English
  • Français

Avoiding Arrays of Odd Order by Latin Squares

Published online by Cambridge University Press:  21 December 2012

LINA J. ANDRÉN ,
CARL JOHAN CASSELGREN  and
LARS-DANIEL ÖHMAN
LINA J. ANDRÉN
Affiliation:
Department of Mathematics and Mathematical Statistics, Umeå University, SE-90187 Umeå, Sweden (e-mail: andren.lina@gmail.com, lars-daniel.ohman@math.umu.se)
CARL JOHAN CASSELGREN
Affiliation:
Department of Mathematics, Linköping University, SE-58183 Linköping, Sweden (e-mail: carl.johan.casselgren@liu.se)
LARS-DANIEL ÖHMAN
Affiliation:
Department of Mathematics and Mathematical Statistics, Umeå University, SE-90187 Umeå, Sweden (e-mail: andren.lina@gmail.com, lars-daniel.ohman@math.umu.se)
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove that there is a constant c such that, for each positive integer k, every (2k + 1) × (2k + 1) array A on the symbols (1,. . .,2k+1) with at most c(2k+1) symbols in every cell, and each symbol repeated at most c(2k+1) times in every row and column is avoidable; that is, there is a (2k+1) × (2k+1) Latin square S on the symbols 1,. . .,2k+1 such that, for each i,j ∈ {1,. . .,2k+1}, the symbol in position (i,j) of S does not appear in the corresponding cell in A. This settles the last open case of a conjecture by Häggkvist. Using this result, we also show that there is a constant ρ, such that, for any positive integer n, if each cell in an n × n array B is assigned a set of m ≤ ρ n symbols, where each set is chosen independently and uniformly at random from {1,. . .,n}, then the probability that B is avoidable tends to 1 as n → ∞.

Keywords

Primary 05B15Secondary 05C15
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 22 , Issue 2 , March 2013 , pp. 184 - 212
Copyright
Copyright © Cambridge University Press 2012

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. H. (2000) The Probabilistic Method, Wiley-Interscience.Google Scholar
[2]Andrén, L. J. (2010) On Latin squares and avoidable arrays. Doctoral thesis, Umeå University.Google Scholar
[3]Asratian, A. S., Denley, T. M. J. and Häggkvist, R. (1998) Bipartite Graphs and their Applications, Cambridge University Press.Google Scholar
[4]Brègman, L. M. (1973) Certain properties of nonnegative matrices and their permanents. Dokl. Akad. Nauk SSSR 211 27–30.Google Scholar
[5]Casselgren, C. J. (2012) On avoiding some families of arrays. Discrete Math. 312 963–972.Google Scholar
[6]Cavenagh, N. J. (2010) Avoidable partial Latin squares of order 4m+1. Ars Combin. 95 257–275.Google Scholar
[7]Chetwynd, A. G. and Rhodes, S. J. (1997) Avoiding multiple entry arrays. J. Graph Theory 25 257–266.Google Scholar
[8]Chetwynd, A. G. and Rhodes, S. J. (1997) Avoiding partial Latin squares and intricacy. Discrete Math. 177 17–32.Google Scholar
[9]Csaba, B. (2007) Regular spanning subgraphs of bipartite graphs of high minimum degree. Electron. J. Combin. 14 N21.Google Scholar
[10]Cutler, J. and Öhman, L.-D. (2006) Latin squares with forbidden entries. Electron. J. Combin. 13 R47.CrossRefGoogle Scholar
[11]Häggkvist, R. (1989) A note on Latin squares with restricted support. Discrete Math. 75 253–254.CrossRefGoogle Scholar
[12]Öhman, L.-D. (2011) Partial Latin squares are avoidable. Ann. Combin. 15 485–497.Google Scholar