In this paper we develop two methods for completing partial latin squares and prove the following. Let $$A$$A be a partial latin square of order $$nr$$nr in which all non-empty cells occur in at most $$n-1$$n-1$$r\times r$$r×r squares. If $$t_1,\ldots , t_m$$t1,?,tm are positive integers for which $$n\geqslant t_1^2+t_2^2+\cdots +t_m^2+1$$n?t12+t22+?+tm2+1 and if $$A$$A is the union of $$m$$m subsquares each with order $$rt_i$$rti, then $$A$$A can be completed. We additionally show that if $$n\geqslant r+1$$n?r+1 and $$A$$A is the union of $$n$$n identical $$r\times r$$r×r squares with disjoint rows and columns, then $$A$$A can be completed. For smaller values of $$n$$n we show that a completion does not always exist.
2Embedding partial Latin squares in Latin squares with many mutually orthogonal mates
Embedding partial Latin squares in Latin squares with many mutually orthogonal mates
9On the spectrum of critical sets in latin squares of order 2n
Suppose that L is a latin square of order m and P ⊑ L is a partial latin square. If L is the only latin square of order m which contains P, and no proper subset of P has this property, then P is a critical set of L. The critical set spectrum problem is to determine, for a given m, the set of integers t for which there exists a latin square of order m with a critical set of size t. We outline a partial solution to the critical set spectrum problem for latin squares of order 2n. The back circulant latin square of even order m has a well‐known critical set of size m2/4, and this is the smallest known critical set for a latin square of order m. The abelian 2‐group of order 2n has a critical set of size 4n‐3n, and this is the largest known critical set for a latin square of order 2n. We construct a set of latin squares with associated critical sets which are intermediate between the back circulant latin square of order 2n and the abelian 2‐group of order 2n. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 25–43, 2008
Read more10Completions of ‐Dense Partial Latin Squares
A classical question in combinatorics is the following: given a partial Latin square P, when can we complete P to a Latin square L? In this paper, we investigate the class of ε‐dense partial Latin squares: partial Latin squares in which each symbol, row, and column contains no more than ‐many nonblank cells. Based on a conjecture of Nash‐Williams, Daykin and Häggkvist conjectured that all ‐dense partial Latin squares are completable. In this paper, we will discuss the proof methods and results used in previous attempts to resolve this conjecture, introduce a novel technique derived from a paper by Jacobson and Matthews on generating random Latin squares, and use this technique to study ε‐dense partial Latin squares that contain no more than filled cells in total. In this paper, we construct completions for all ε‐dense partial Latin squares containing no more than filled cells in total, given that . In particular, we show that all ‐dense partial Latin squares are completable. These results improve prior work by Gustavsson, which required , as well as Chetwynd and Häggkvist, which required , n even and greater than 107.
Read more19Avoiding partial Latin squares and intricacy
Avoiding partial Latin squares and intricacy
1A massively parallel evolutionary algorithm for the partial Latin square extension problem
A massively parallel evolutionary algorithm for the partial Latin square extension problem
253-Homogeneous latin trades
3-Homogeneous latin trades
6Enumeration of Isotopy Classes of Diagonal Latin Squares of Small Order Using Volunteer Computing
The paper is devoted to discovering new features of diagonal Latin squares of small order. We present an algorithm, based on a special kind of transformations, that constructs a canonical form of a given diagonal Latin square. Each canonical form corresponds to one isotopy class of diagonal Latin squares. The algorithm was implemented and used to enumerate the isotopy classes of diagonal Latin squares of order at most 8. For order 8 the computational experiment was conducted in a volunteer computing project. The algorithm was also used to estimate how long it would take to enumerate the isotopy classes of diagonal Latin squares of order 9 in the same volunteer computing project.
Read moreAvoiding Pairs of Partial Latin Squares
We show that two partial latin squares of order mk are simultaneously avoidable if m > 4 and $${k>\frac{m^3(m^2-1)}{2}}$$ . If m = 4, we show the same conclusion when k > 56.
Read more6Iterated local search with Trellis-neighborhood for the partial Latin square extension problem
A partial Latin square (PLS) is a partial assignment of n symbols to an $$n\times n$$n×n grid such that, in each row and in each column, each symbol appears at most once. The PLS extension problem is an NP-hard problem that asks for a largest extension of a given PLS. We consider the local search such that the neighborhood is defined by (p, q)-swap , i.e., the operation of dropping exactly p symbols and then assigning symbols to at most q empty cells. As a fundamental result, we provide an efficient $$(p,\infty )$$(p,ź)-neighborhood search algorithm that finds an improved solution or concludes that no such solution exists for $$p\in \{1,2,3\}$$pź{1,2,3}. The running time of the algorithm is $$O(n^{p+1})$$O(np+1). We then propose a novel swap operation, Trellis-swap, which is a generalization of (p, q)-swap with $$p\le 2$$p≤2. The proposed Trellis-neighborhood search algorithm runs in $$O(n^{3.5})$$O(n3.5) time. The iterated local search (ILS) algorithm with Trellis-neighborhood is more likely to deliver a high-quality solution than not only ILSs with $$(p,\infty )$$(p,ź)-neighborhood but also state-of-the-art optimization solvers such as IBM ILOG CPLEX and LocalSolver.
Read more7Autotopism stabilized colouring games on rook’s graphs
Autotopism stabilized colouring games on rook’s graphs
20Indivisible partitions of latin squares
Indivisible partitions of latin squares
27On the finite completion of partial latin cubes
On the finite completion of partial latin cubes
5On the maximality of a set of mutually orthogonal Sudoku Latin Squares
The maximum number of mutually orthogonal Sudoku Latin squares (MOSLS) of order $$n=m^2$$n=m2 is $$n-m$$n-m. In this paper, we construct for $$n=q^2$$n=q2, q a prime power, a set of $$q^2-q-1$$q2-q-1 MOSLS of order $$q^2$$q2 that cannot be extended to a set of $$q^2-q$$q2-q MOSLS. This contrasts to the theory of ordinary Latin squares of order n, where each set of $$n-2$$n-2 mutually orthogonal Latin Squares (MOLS) can be extended to a set of $$n-1$$n-1 MOLS (which is best possible). For this proof, we construct a particular maximal partial spread of size $$q^2-q+1$$q2-q+1 in $$\mathrm {PG}(3,q)$$PG(3,q) and use a connection between Sudoku Latin squares and projective geometry, established by Bailey, Cameron and Connelly.
Read more5Enumeration of Latin squares with conjugate symmetry
A Latin square has six conjugate Latin squares obtained by uniformly permuting its (row, column, symbol) triples. We say that a Latin square has conjugate symmetry if at least two of its six conjugates are equal. We enumerate Latin squares with conjugate symmetry and classify them according to several common notions of equivalence. We also do similar enumerations under additional hypotheses, such as assuming the Latin square is reduced, diagonal, idempotent or unipotent. Our data corrected an error in earlier literature and suggested several patterns that we then found proofs for, including (1) the number of isomorphism classes of semisymmetric idempotent Latin squares of order equals the number of isomorphism classes of semisymmetric unipotent Latin squares of order , and (2) suppose and are totally symmetric Latin squares of order . If and are paratopic then and are isomorphic.
Read more19Partial Latin rectangle graphs and autoparatopism groups of partial Latin rectangles with trivial autotopism groups
An r×s partial Latin rectangle (lij) is an r×s matrix containing elements of {1,2,…,n}∪{⋅} such that each row and each column contain at most one copy of any symbol in {1,2,…,n}. An entry is a triple (i,j,lij) with lij≠⋅. Partial Latin rectangles are operated on by permuting the rows, columns, and symbols, and by uniformly permuting the coordinates of the set of entries. The stabilizers under these operations are called the autotopism group and the autoparatopism group, respectively.We develop the theory of symmetries of partial Latin rectangles, introducing the concept of a partial Latin rectangle graph. We give constructions of m-entry partial Latin rectangles with trivial autotopism groups for all possible autoparatopism groups (up to isomorphism) when: (a) r=s=n, i.e., partial Latin squares, (b) r=2 and s=n, and (c) r=2 and s≠n.
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