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 if BðA; skÞ is a cycle of length 4n for k 1⁄4 1;y; n: Two double latin squares ðaijÞ; ðaijÞ of order 2n on symbols s1;y; sn are said to be orthogonal if for each ordered pair ðsh; skÞ of symbols there are four ordered pairs ði; jÞ such that aij 1⁄4 sh; aij 1⁄4 sk: 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 *Corresponding author. E-mail address: [email protected] (A.J.W. Hilton). This research is supported by ONR Grant N00014-95-10769 and Grant DMS-9531722. Sadly Crispin Nash-Williams died while this version of this paper was being written. 0095-8956/02/$ see front matter r 2002 Elsevier Science (USA). All rights reserved. PII: S 0 0 9 5 8 9 5 6 ( 0 2 ) 0 0 0 2 9 1 equivalent to decompositions of K2n into Hamiltonian paths (and these are equivalent to decompositions of K2nþ1 into Hamiltonian cycles). r 2002 Elsevier Science (USA). All rights reserved. 1. Definition and elementary construction A double latin square of order 2n is a 2n 2n matrix containing n symbols, such that each cell contains exactly one symbol and each symbol occurs exactly twice in each row and twice in each column. The occurrences of a symbol s describe a set of disjoint cycles in a double latin square: if s occurs in 2n distinct cells ði1; j1Þ; ði1; j2Þ; ði2; j2Þ; ði2; j3Þ; ði3; j3Þ; ði3; j4Þ;y; ðic; jcÞ; ðic; j1Þ then these cells are said to constitute a cycle, or more specifically a s-cycle, of length 2c: In a double latin square of order 2n; the lengths of the cycles described by any one symbol have sum 4n: A cycle of length 4n; the maximum possible length, is called a Hamiltonian cycle of the double latin square. In this paper we study double latin squares in which the occurrences of each symbol describe a Hamiltonian cycle. Such double latin squares are called Hamiltonian double latin squares. The expression ‘‘Hamiltonian double latin square(s) of order 2n’’ will be abbreviated to HLSð2nÞ: We let Aði; jÞ denote the entry in the cell ði; jÞ of a matrix A: If A is an n n matrix and g is a permutation of the set f1;y; ng then pgðAÞ will denote the matrix obtained from A by applying the permutation g to its columns and pgðAÞ will denote the matrix obtained from A by applying the permutation g to its rows: thus pgðAÞ 1⁄4 B; pgðAÞ 1⁄4 C where Bði; gðjÞÞ 1⁄4 CðgðiÞ; jÞ 1⁄4 Aði; jÞ for i; j 1⁄4 1;y; n: The following theorem (which incorporates an improvement suggested by a referee) describes an easy way to construct several HLSð2nÞ from two latin squares of order n: Theorem 1.1. If A;B are latin squares of order n on the same n symbols and g is a permutation of f1;y; ng which has just one cycle (i.e. 1; gð1Þ; gð1Þ; gð1Þ;y; g ð1Þ are distinct) then