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# Completions of ε-Dense Partial Latin Squares

## Citation

Bartlett, Padraic James (2013) Completions of ε-Dense Partial Latin Squares. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/GQZE-RB50. https://resolver.caltech.edu/CaltechTHESIS:06032013-015803040

## 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 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.

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\$.

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\$.

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}\$.

This strengthens results of Gustavsson, which requires \$epsilon = delta = 10^{-7}\$.

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.

Item Type: Thesis (Dissertation (Ph.D.)) Latin squares, partial Latin squares, quasirandom graphs California Institute of Technology Physics, Mathematics and Astronomy Mathematics Public (worldwide access) Wilson, Richard M. Wilson, Richard M. (chair)Wales, David B.Kechris, Alexander S.Omar, Mohamed 28 May 2013 CaltechTHESIS:06032013-015803040 https://resolver.caltech.edu/CaltechTHESIS:06032013-015803040 10.7907/GQZE-RB50 No commercial reproduction, distribution, display or performance rights in this work are provided. 7819 CaltechTHESIS Padraic Bartlett 06 Jun 2013 19:00 04 Oct 2019 00:01

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