A Generalisation of Transversals for Latin Squares

Abstract

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 $1\leq c < k$.

Latin squares with no small odd plexes

“…Although the concept is much older, the name k-plex was first used in [21]. The definition given in Section 1 is a restriction of the original one.…”

“…The definition given in Section 1 is a restriction of the original one. In defining a k-plex as a partial Latin square, Wanless [21] allowed that a k-plex be not necessarily completable to a Latin square of the same order. Herein, we are solely concerned with k-plexes which are contained in a Latin square.…”

“…Statistical literature sometimes refers to a transversal as a directrix and uses the terms duplex, triplex, and quadruplex for a 2-plex, 3-plex, and 4-plex, respectively. Other names are mentioned in [21].…”

“…Drawing upon the works of Finney [6][7][8], Freeman [10][11][12], Johnston and Fullerton [15], Killgrove et al [17], and Wanless [21], we are able to summarize in Table I the Latin squares of order n ≤ 8 by the named plex ranges. The 34 species of order n ≤ 8 which have a mixed plex range all contain squares with a 3-plex but no transversal.…”

“…The 34 species of order n ≤ 8 which have a mixed plex range all contain squares with a 3-plex but no transversal. It is conjectured [21] that a Latin square with this property exists for all even orders n > 4.…”