Restricted completion of sparse partial Latin squares

Ann×npartial Latin squarePis calledα-dense if each row and column has at mostαnnon-empty cells and each symbol occurs at mostαntimes inP. Ann×narrayAwhere each cell contains a subset of {1,…,n} is a (βn,βn, βn)-array if each symbol occurs at mostβntimes in each row and column and each cell contains a set of size at mostβn. Combining the notions of completing partial Latin squares and avoiding arrays, we prove that there are constantsα,β> 0 such that, for every positive integern, ifPis anα-densen×npartial Latin square,Ais ann×n (βn, βn, βn)-array, and no cell ofPcontains a symbol that appears in the corresponding cell ofA, then there is a completion ofPthat avoidsA; that is, there is a Latin squareLthat agrees withPon every non-empty cell ofP, and, for eachi,jsatisfying 1 ≤i,j≤n, the symbol in position (i,j) inLdoes not appear in the corresponding cell ofA.

Completing Partial Latin Squares with One Nonempty Row, Column, and Symbol

Let $r,c,s\in\{1,2,\ldots,n\}$ and let $P$ be a partial latin square of order $n$ in which each nonempty cell lies in row $r$, column $c$, or contains symbol $s$. We show that if $n\notin\{3,4,5\}$ and row $r$, column $c$, and symbol $s$ can be completed in $P$, then a completion of $P$ exists. As a consequence, this proves a conjecture made by Casselgren and Häggkvist. Furthermore, we show exactly when row $r$, column $c$, and symbol $s$ can be completed.

Completing Partial Latin Squares with Two Filled Rows and Two Filled Columns

It is shown that any partial Latin square of order at least six which consists of two filled rows and two filled columns can be completed.

A Generalisation of Transversals for Latin Squares

We define a $k$-plex to be a partial latin square of order $n$ containing $kn$ entries such that exactly $k$ entries lie in each row and column and each of $n$ symbols occurs exactly $k$ times. A transversal of a latin square corresponds to the case $k=1$. For $k>n/4$ we prove that not all $k$-plexes are completable to latin squares. Certain latin squares, including the Cayley tables of many groups, are shown to contain no $(2c+1)$-plex for any integer $c$. However, Cayley tables of soluble groups have a $2c$-plex for each possible $c$. We conjecture that this is true for all latin squares and confirm this for orders $n\leq8$. Finally, we demonstrate the existence of indivisible $k$-plexes, meaning that they contain no $c$-plex for $1\leq c < k$.

A generalization of Evans' Theorem: embedding partial tricycle systems

In 1960, Trevor Evans gave a best possible embedding of a partial latin square of order n in a latin square of order t, for any t ≥ 2n. A latin square of order n is equivalent to a 3-cycle system of Kn, n, n, the complete tripartite graph. Here we consider a small embedding of partial 3k-cycle systems of Kn, n, n of a certain type which generalizes Evans' Theorem, and discuss how this relates to the embedding of patterned holes, another recent generalization of Evans' Theorem.