Restricted completion of sparse partial Latin squares

TL;DR

This paper proves that under certain density constraints, it is possible to complete partial Latin squares while avoiding specified arrays of forbidden symbols, extending the understanding of Latin square completions with restrictions.

Contribution

The authors establish the existence of completions for dense partial Latin squares that avoid given arrays of forbidden symbols, combining partial Latin square completion with avoidance constraints.

Findings

Existence of completions under density conditions

Completion avoids specified arrays of symbols

Applicable for all sufficiently large n

Abstract

An $n \times n$ partial Latin square $P$ is called $\alpha$-dense if each row and column has at most $\alpha n$ non-empty cells and each symbol occurs at most $\alpha n$ times in $P$. An $n \times n$ array $A$ where each cell contains a subset of $\{1,\dots, n\}$ is a $(\beta n, \beta n, \beta n)$-array if each symbol occurs at most $\beta n$ times in each row and column and each cell contains a set of size at most $\beta n$. Combining the notions of completing partial Latin squares and avoiding arrays, we prove that there are constants $\alpha, \beta > 0$ such that, for every positive integer $n$, if $P$ is an $\alpha$-dense $n \times n$ partial Latin square, $A$ is an $n \times n$ $(\beta n, \beta n, \beta n)$-array, and no cell of $P$ contains a symbol that appears in the corresponding cell of $A$, then there is a completion of $P$ that avoids $A$; that is, there is a Latin square $L$ that agrees with $P$ on every non-empty cell of $P$, and, for each $i,j$ satisfying $1 \leq i,j \leq n$, the symbol in position $(i,j)$ in $L$ does not appear in the corresponding cell of $A$.