Completions of ‐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.