Generalized Latin squares and their defining sets

Abstract

A generalized Latin square of type ( n , k ) is an n × n array of symbols 1 , 2 , … , k such that each of these symbols occurs at most once in each row and each column. Let d ( n , k ) denote the cardinality of the minimal set S of given entries of an n × n array such that there exists a unique extension of S to a generalized Latin square of type ( n , k ) . In this paper we discuss the properties of d ( n , k ) for k = 2 n - 1 and k = 2 n - 2 . We give an alternate proof of the identity d ( n , 2 n - 1 ) = n 2 - n , which holds for even n , and we establish the new result d ( n , 2 n - 2 ) ⩾ n 2 - ⌊ 8 n 5 ⌋ . We also show that the latter bound is tight for n divisible by 10.