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Completions of epsilon-dense partial Latin squares; quasirandom k-colorings of graphs

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 \textbf{$ε$-dense partial Latin squares}:\ partial Latin squares in which each symbol, row, and column contains no more than $εn$-many nonblank cells. Based on a conjecture of Nash-Williams, Daykin and Häggkvist conjectured that all $\frac{1}{4}$-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 novel technique to study $ ε$-dense partial Latin squares that contain no more than $δn^2$ filled cells in total. In particular, we construct completions for all $ ε$-dense partial Latin squares containing no more than $δn^2$ filled cells in total, given that $ε< \frac{1}{12}, δ< \frac{ \left(1-12ε\right)^{2}}{10409}$. In particular, we show that all $9.8 \cdot 10^{-5}$-dense partial Latin squares are completable. We further show that such completions can always be found in polynomial time. This contrasts a result of Colbourn. In Chapter 3, we strengthen Colbourn's result to the claim that completing an arbitrary $\left(\frac{1}{2} + ε\right)$-dense partial Latin square is NP-complete, for any $ε> 0$. Additional results on triangulations of graphs are found. In an unrelated vein, Chapter 6 explores the class of quasirandom graphs. In specific, we study quasirandom $k$-edge colorings, and create an analogue of Chung, Graham and Wilson's well-known results for such colorings.