Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-26T01:16:10.984Z Has data issue: false hasContentIssue false
  • English
  • Français

Latin Squares with Prescribed Diagonals

Published online by Cambridge University Press:  20 November 2018

A. J. W. Hilton  and
C. A. Rodger
A. J. W. Hilton
Affiliation:
University of Reading, Reading, England
C. A. Rodger
Affiliation:
University of Reading, Reading, England
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

1. Introduction. An incomplete latin rectangle on t symbols σ1 …, σt of size r × s is an r × s matrix in which each cell is occupied by exactly one of the symbols σ1 …, σt in such a way that no symbol occurs more than once in any row or more than once in any column. If r = t or s = t then it is a latin rectangle; if r = s < t it is an incomplete latin square; if r = s = t it is a latin square. The diagonal of a latin square consists of the cells (i, i) (1 ≦ i ≦ t) together with the symbols occupying those cells. Let an allowed sequence of length t be a sequence of length t in which no symbol occurs exactly t – 1 times. Let an allowed diagonal of length t be a diagonal occupied by an allowed sequence.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 34 , Issue 6 , 01 December 1982 , pp. 1251 - 1254
Copyright
Copyright © Canadian Mathematical Society 1982

References

1. Andersen, L. D., Latin squares and their generalizations, Ph.D. Thesis, University of Reading (1979).Google Scholar
2. Andersen, L. D., Embedding latin squares with prescribed diagonal, Annals of Discrete Math. 15 (1982), 9–26.Google Scholar
3. Andersen, L. D. and A. J. W, Hilton, Thank Evans!, to appear.Google Scholar
4. Andersen, L. D., A. J. W, Hilton and Rodger, C. A., A solution to the embedding problem for partial idempotent latin squares, J. London Math. Soc, to appear.Google Scholar
5. Chang, G. J., Complete diagonals of latin squares, Can. Math. Bull. 22 (1979), 477–481.Google Scholar
6. Hall, M. Jr., An existence theorem for latin squares, Bull. Amer. Math. Soc. 51 (1945), 387–388.Google Scholar
7. Hilton, A. J. W., Embedding incomplete latin rectangles, Annals of Discrete Math. 13 (1982), 121–138.Google Scholar