On avoiding some families of arrays

An n × n array A with entries from { 1 , … , n } is avoidable if there is an n × n Latin square L such that no cell in L contains a symbol that occurs in the corresponding cell in A . We show that the problem of determining whether an array that contains at most two entries per cell is avoidable is NP -complete, even in the case when the array has entries from only two distinct symbols. Assuming that P ≠ NP , this disproves a conjecture by Öhman. Furthermore, we present several new families of avoidable arrays. In particular, every single entry array (arrays where each cell contains at most one symbol) of order n ≥ 2 k with entries from at most k distinct symbols and where each symbol occurs in at most n − 2 cells is avoidable, and every single entry array of order n , where each of the symbols 1 , … , n occurs in at most ⌊ n 5 ⌋ cells, is avoidable. Additionally, if k ≥ 2 , then every single entry array of order at least n ≥ 4 , where at most k rows contain non-empty cells and where each symbol occurs in at most n − k + 1 cells, is avoidable.