Hamiltonian double latin squares

Hamiltonian double latin squares

A double latin square of order 2n on symbols s1;y; sn is a 2n 2n matrix A 1⁄4 ðaijÞ in which each aij is one of the symbols s1;y; sn and each sk occurs twice in each row and twice in each column. For k 1⁄4 1;y; n let BðA; skÞ be the bipartite graph with vertices r1;y; r2n; c1;y; c2n and 4n edges 1⁄2ri; cj corresponding to ordered pairs ði; jÞ such that aij 1⁄4 sk: We say that A is Hamiltonian i...

A method for constructing symmetric Hamiltonian double Latin squares

Lifting Redundancy from Latin Squares to Pandiagonal Latin Squares

In the pandiagonal Latin Square problem, a square grid of size N needs to be filled with N types of objects, so that each column, row, and wrapped around diagonal (both up and down) contains an object of each type. This problem dates back to at least Euler. In its specification as a constraint satisfaction problem, one uses the all different constraint. The known redundancy result about all dif...

Latin Squares: Transversals and counting of Latin squares

Author: Jenny Zhang First, let’s preview what mutually orthogonal Latin squares are. Two Latin squares L1 = [aij ] and L2 = [bij ] on symbols {1, 2, ...n}, are said to be orthogonal if every ordered pair of symbols occurs exactly once among the n2 pairs (aij , bij), 1 ≤ i ≤ n, 1 ≤ j ≤ n. Now, let me introduce a related concept which is called transversal. A transversal of a Latin square is a se...

Amalgamating infinite latin squares

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...