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Keywords:

  • Latin squares;
  • partial Latin squares;
  • Häggkvist;
  • Gustavsson;
  • 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 inline image-many nonblank cells. Based on a conjecture of Nash-Williams, Daykin and Häggkvist conjectured that all inline image-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 inline image filled cells in total. In this paper, we construct completions for all ε-dense partial Latin squares containing no more than inline image filled cells in total, given that inline image. In particular, we show that all inline image-dense partial Latin squares are completable. These results improve prior work by Gustavsson, which required inline image, as well as Chetwynd and Häggkvist, which required inline image, n even and greater than 107.