A diagonal Latin square is a Latin square whose main diagonal and back diagonal are both transversals. A Latin square is self-orthogonal if it is orthogonal to its transpose. In an earlier paper Danhof, Phillips and Wallis considered the question of the existence of self-orthogonal diagonal Latin squares of order 10. In this paper we shall present some constructions of self-orthogonal diagonal ...

The chromatic number of a Latin square is the least number of partial transversals which cover its cells. This is just the chromatic number of its associated Latin square graph. Although Latin square graphs have been widely studied as strongly regular graphs, their chromatic numbers appear to be unexplored. We determine the chromatic number of a circulant Latin square, and find bounds for some ...

A diagonal is said to be complete if every element appears in it exactly once. For n = 2m even, we introduce the concept of a crisscross Latin square which is something in between a diagonal Latin square and a Knut Vik design. A crisscross Latin square is a Latin square such that all the jth right diagonals for even j and all the jth left diagonals for odd j are complete. We show that a necessa...

A finite latin square is an n × n matrix whose entries are elements of the set {1, . . . , n} and no element is repeated in any row or column. Given equivalence relations on the set of rows, the set of columns, and the set of symbols, respectively, we can use these relations to identify equivalent rows, columns and symbols, and obtain an amalgamated latin square. There is a set of natural equat...

A partial latin square on t symbols <rl5..., ut of side n is an n x n matrix, each of the cells of which may be empty or may be occupied by. one of the symbols ol,..., at, and which satisfies the rule that no symbol occurs more than once in any row or more than once in any column. An incomplete latin square on t symbols of side n is a partial latin square in which there are no empty cells; it i...

This paper first reviews some basic properties of cryptographic hash function, secret sharing scheme, and Latin square. Then we discuss why Latin square or its critical set is a good choice for secret representation and its relationship with secret sharing scheme. Further we enumerate the limitations of Latin square in a secret sharing scheme. Finally we propose how to apply cryptographic hash ...

The problem of completing partial latin squares to latin squares of the same order has been studied for many years. For instance, in 1960 Evans [9] conjectured that every partial latin square of order n containing at most n− 1 filled cells is completable to a latin square of order n. This conjecture was shown to be true by Lindner [12] and Smetaniuk [13]. Recently, Bryant and Rodger [6] establi...