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A Caltech Library Repository Feedhttp://www.rssboard.org/rss-specificationpython-feedgenenWed, 31 Jan 2024 18:49:22 +0000Completions of ε-Dense Partial Latin Squares
https://resolver.caltech.edu/CaltechTHESIS:06032013-015803040
Authors: {'items': [{'email': 'chionophilia@gmail.com', 'id': 'Bartlett-Padraic-James', 'name': {'family': 'Bartlett', 'given': 'Padraic James'}, 'show_email': 'NO'}]}
Year: 2013
DOI: 10.7907/GQZE-RB50
<p>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 H"aggkvist 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. </p>
<p>In Chapter 2, 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. In Chapter 4, we augment these results by roughly a factor of two using some probabilistic techniques. These results improve prior work by Gustavsson, which required $epsilon = delta leq 10^{-7}$, as well as Chetwynd and H"aggkvist, which required $epsilon = delta = 10^{-5}$, $n$ even and greater than $10^7$.</p>
<p>If we omit the probabilistic techniques noted above, we further show that such completions can always be found in polynomial time. This contrasts a result of Colbourn, which states that completing arbitrary partial Latin squares is an NP-complete task. 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 $epsilon > 0$.</p>
<p>Colbourn's result hinges heavily on a connection between triangulations of tripartite graphs and Latin squares. Motivated by this, we use our results on Latin squares to prove that any tripartite graph $G = (V_1, V_2, V_3)$ such that
begin{itemize}
item $|V_1| = |V_2| = |V_3| = n$,
item For every vertex $v in V_i$, $deg_+(v) = deg_-(v) geq (1- epsilon)n,$ and
item $|E(G)| > (1 - delta)cdot 3n^2$
end{itemize}
admits a triangulation, if $epsilon < frac{1}{132}$, $delta < frac{(1 -132epsilon)^2 }{83272}$. In particular, this holds when $epsilon = delta=1.197 cdot 10^{-5}$.</p>
<p>This strengthens results of Gustavsson, which requires $epsilon = delta = 10^{-7}$. </p>
<p>In an unrelated vein, Chapter 6 explores the class of textbf{quasirandom graphs}, a notion first introduced by Chung, Graham and Wilson cite{chung1989quasi} in 1989. Roughly speaking, a sequence of graphs is called "quasirandom"' if it has a number of properties possessed by the random graph, all of which turn out to be equivalent. In this chapter, we study possible extensions of these results to random $k$-edge colorings, and create an analogue of Chung, Graham and Wilson's result for such colorings.</p>https://thesis.library.caltech.edu/id/eprint/7819