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- W1500742022 abstract "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{$epsilon$-dense partial Latin squares}: partial Latin squares in which each symbol, row, and column contains no more than $epsilon n$-many nonblank cells. Based on a conjecture of Nash-Williams, Daykin and Haggkvist 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 $ epsilon$-dense partial Latin squares that contain no more than $delta n^2$ filled cells in total. In particular, we construct completions for all $ epsilon$-dense partial Latin squares containing no more than $delta n^2$ filled cells in total, given that $epsilon < frac{1}{12}, delta < frac{ left(1-12epsilonright)^{2}}{10409}$. 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 $left(frac{1}{2} + epsilonright)$-dense partial Latin square is NP-complete, for any $epsilon > 0$. 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." @default.
- W1500742022 created "2016-06-24" @default.
- W1500742022 creator A5040736324 @default.
- W1500742022 date "2013-06-03" @default.
- W1500742022 modified "2023-09-27" @default.
- W1500742022 title "Completions of epsilon-dense partial Latin squares; quasirandom k-colorings of graphs" @default.
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