Completions of ε-Dense Partial Latin Squares
Article first published online: 10 JUN 2013
© 2013 Wiley Periodicals, Inc.
Journal of Combinatorial Designs
Volume 21, Issue 10, pages 447–463, October 2013
How to Cite
Bartlett, P. (2013), Completions of ε-Dense Partial Latin Squares. J. Combin. Designs, 21: 447–463. doi: 10.1002/jcd.21355
- Issue published online: 1 AUG 2013
- Article first published online: 10 JUN 2013
- Manuscript Revised: 9 MAY 2013
- Manuscript Received: 7 MAY 2012
- Latin squares;
- partial Latin squares;
- improper Latin squares;
- epsilon-dense partial Latin squares
A classical question in combinatorics is the following: given a partial Latin square P, when can we complete P to a Latin square L? In this paper, we investigate the class of ε-dense partial Latin squares: partial Latin squares in which each symbol, row, and column contains no more than -many nonblank cells. Based on a conjecture of Nash-Williams, Daykin and Häggkvist conjectured that all -dense partial Latin squares are completable. In this paper, we will discuss the proof methods and results used in previous attempts to resolve this conjecture, introduce a novel technique derived from a paper by Jacobson and Matthews on generating random Latin squares, and use this technique to study ε-dense partial Latin squares that contain no more than filled cells in total. In this paper, we construct completions for all ε-dense partial Latin squares containing no more than filled cells in total, given that . In particular, we show that all -dense partial Latin squares are completable. These results improve prior work by Gustavsson, which required , as well as Chetwynd and Häggkvist, which required , n even and greater than 107.