Latin trades on three or four rows

Abstract

Latin trades are closely related to the problem of critical sets in Latin squares. We denote the cardinality of the smallest critical set in any Latin square of order n by scs ( n ) . A consideration of Latin trades which consist of just two columns, two rows, or two elements establishes that scs ( n ) ⩾ n - 1 . We conjecture that a consideration of Latin trades on four rows may establish that scs ( n ) ⩾ 2 n - 4 . We look at various attempts to prove a conjecture of Cavenagh about such trades. The conjecture is proven computationally for values of n less than or equal to 9. In particular, we look at Latin squares based on the group table of Z n for small n and trades in three consecutive rows of such Latin squares.