Hamiltonian double latin squares

Abstract

A double latin square of order 2 n on symbols σ 1,…, σ n is a 2 n×2 n matrix A=( a ij ) in which each a ij is one of the symbols σ 1,…, σ n and each σ k occurs twice in each row and twice in each column. For k=1,…, n let B( A, σ k ) be the bipartite graph with vertices ρ 1,…, ρ 2 n , c 1,…, c 2 n and 4 n edges [ ρ i , c j ] corresponding to ordered pairs ( i, j) such that a ij = σ k . We say that A is Hamiltonian if B( A, σ k ) is a cycle of length 4 n for k=1,…, n. Two double latin squares ( a ij ),( a ij ′) of order 2 n on symbols σ 1,…, σ n are said to be orthogonal if for each ordered pair ( σ h , σ k ) of symbols there are four ordered pairs ( i, j) such that a ij = σ h , a ij ′= σ k . We explore ways of constructing Hamiltonian double latin squares (HLS), symmetric HLS, sets of mutually orthogonal HLS and pairs of orthogonal symmetric HLS. We identify those arrays which can be obtained from HLS by amalgamating rows and amalgamating columns in a certain sense, and we prove a similar result concerning symmetric arrays obtainable in this way from symmetric HLS. These results can be proved either by using matroids or by a more elementary method, and we illustrate both approaches. From these results we deduce a characterisation of those matrices which are submatrices of HLS on n symbols, a similar result concerning symmetric submatrices of symmetric HLS and some related results. Much of our discussion uses graph-theoretic language, since HLS on n symbols are equivalent to decompositions of K 2 n,2 n into Hamiltonian cycles and symmetric HLS on n symbols are equivalent to decompositions of K 2 n into Hamiltonian paths (and these are equivalent to decompositions of K 2 n+1 into Hamiltonian cycles).