A diagonal Latin square is a Latin square whose main diagonal and back diagonal are both transversals. In this paper we give some constructions of pairwise orthogonal diagonal Latin squares. As an application of such constructions we obtain some new infinite classes of pairwise orthogonal diagonal Latin squares which are useful in the study of pairwise orthogonal diagonal Latin squares.

A Latin square is pan-Hamiltonian if the permutation which defines row i relative to row j consists of a single cycle for every i 61⁄4 j. A Latin square is atomic if all of its conjugates are pan-Hamiltonian. We give a complete enumeration of atomic squares for order 11, the smallest order for which there are examples distinct from the cyclic group. We find that there are seven main classes, in...

A latin square is an n x n square matrix each of which cells contains a symbol chosen from the set 11,2, ... ,n]; each symbol occurs exactly oncl~ in each row or column of the matrix. A partial latin square is a latin square in which some cells are unoccupied. We consider the problem of obtaining necessary and sufficient conditions for a partial latin square to be completed to a latin square. F...

A Latin square design whose automorphism group is transitive of rank at most 3 on points must come from the multiplication table of an elementary abelian p-group,

In this paper, we introduce a symmetric-key Latin square image cipher (LSIC) for grayscale and color images. Our contributions to the image encryption community include 1) we develop new Latin square image encryption primitives including Latin Square Whitening, Latin Square S-box and Latin Square P-box ; 2) we provide a new way of integrating probabilistic encryption in image encryption by embe...

A critical set in a latin square is a subset of its elements with the following properties: 1) No other latin square exists which also contains that subset. 2) No element may be deleted without destroying property 1. Let scs(n) denote the smallest possible cardinality of a critical set in an n × n latin square. It is conjectured that scs(n) = n/4 , and that only the back-circulant latin square ...

In combinatorics, a latin square is $$n\times n$$ matrix filled with n different symbols, each occurring exactly once in row and column. Associated to square, we can define simple graph called graph. this article, compute lower upper bounds for the domination number k-tuple total numbers of such graphs. Moreover, describe formula 2-tuple number.