Critical Sets in Latin Squares and Associated Structures

Abstract

A critical set in a Latin square of order $n$ is a set of entries in an $n\times n$ array which can be embedded in precisely one Latin square of order $n$, with the property that if any entry of the critical set is deleted, the remaining set can be embedded in more than one Latin square of order $n$. The cardinality of the largest critical set in any Latin square of order $n$ is denoted by $lcs(n)$. In 1978 Curran and van Rees proved that $lcs(n)\leq n^2-n$. In Chapter 4, it is shown that $lcs(n)\leq n^2-3n+3$. Chapter 5 provides new bounds on the maximum number of intercalates in Latin squares of orders $2^\alpha m$ and $2^\alpha m+1$, and a new lower bound on $lcs(4m)$. In Chapter 6 a construction is given which verifies the existence of a critical set of size $\displaystyle{\frac{n^2}{4}} + 1$ when $n$ is even and $n\geq 6$. In Chapter 7 the representation of Steiner trades of volume less than or equal to nine is examined. Computational results are used to identify those trades for which the associated partial Latin square can be decomposed into six disjoint Latin interchanges. Chapter 8 focusses on critical sets in Latin squares of order at most six and extensive computational routines are used to identify all the critical sets of different sizes in these Latin squares.