Hamiltonian double Latin squares

  • Authors:

  • A. J. W. Hilton;M. Mays;C. A. Rodger;C. St. J. A. Nash-Williams

  • Affiliations:

  • Department of Mathematics, The University of Reading, Whiteknights, P.O. Box 220, Reading RG6 6AX, UK;Department of Mathematics, West Virginia University, Morgantown, P.O. Box 6310, WV;Department of Discrete and Statistical Sciences, 120 Mathematics Annex, Auburn University, AL;Department of Mathematics, The University of Reading, Whiteknights, P.O. Box 220, Reading RG6 6AX, UK

  • Venue:
  • Journal of Combinatorial Theory Series B
  • Year:
  • 2003

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Abstract

A double latin square of order 2n on symbols σ1, ..., σn an is a 2n × 2n matrix A = (aij) in which each aij is one of the symbols σ1, ..., σn and each σk occurs twice in each row and twice in each column. For k = 1,...,n let B(A, σk) be the bipartite graph with vertices ρ1,..., ρ2n, c1, ..., c2n and 4n edges [ρi, cj] corresponding to ordered pairs (i,j) such that aij = σk. We say that A is Hamiltonian if B(A, σk) is a cycle of length 4n for k = 1 ,..., n. Two double latin squares (aij), (a'ij) of order 2n on symbols σ1,...,σn are said to be orthogonal if for each ordered pair (σh, σk) of symbols there are four ordered pairs (i,j) such that aij = σh, a'ij = σk.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 K2n,2n into Hamiltonian cycles and symmetric HLS on n symbols are equivalent to decompositions of K2n into Hamiltonian paths (and these are equivalent to decompositions of K2n+1 into Hamiltonian cycles).