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

TL;DR

This paper advances understanding of partial Latin square completion by introducing new techniques to prove that very dense partial Latin squares are completable in polynomial time, and explores quasirandom graph colorings.

Contribution

It introduces a novel technique for completing epsilon-dense partial Latin squares and establishes polynomial-time algorithms for certain densities, contrasting with NP-completeness results for higher densities.

Findings

All 9.8×10^{-5}-dense partial Latin squares are completable.

Polynomial-time algorithms exist for completing these dense Latin squares.

Completing (1/2 + epsilon)-dense partial Latin squares is NP-complete.

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 particular, we construct completions for all $ \epsilon$-dense partial Latin squares containing no more than $\delta n^2$ filled cells in total, given that $\epsilon < \frac{1}{12}, \delta < \frac{ \left(1-12\epsilon\right)^{2}}{10409}$. 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 $\left(\frac{1}{2} + \epsilon\right)$-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.