Completing partial latin squares with one nonempty row, column, and symbol

Let r,c,s ∈{1,2,…,n} and let PP be a partial latin square of order n in which each nonempty cell lies in row r, column c, or contains symbol s. We show that if n ∉ {3, 4, 5} and row r, column c, and symbol s can be completed in P, then a completion of P exists. As a consequence, this proves a conjecture made by Casselgren and Häggkvist. Furthermore, we show exactly when row r, column c, and symbol s can be completed

Induced subarrays of Latin squares without repeated symbols

We show that for any Latin square L of order 2m, we can partition the rows and columns of L into pairs so that at most (m+3)/2 of the 2x2 subarrays induced contain a repeated symbol. We conjecture that any Latin square of order 2m (where m ≥ 2, with exactly five transposition class exceptions of order 6) has such a partition so that every 2x2 subarray induced contains no repeated symbol. We verify this conjecture by computer when m ≤ 4

On the Existence of Partitioned Incomplete Latin Squares with Five Parts

Let a, b, c, d, e be positive integers such that a ≤ b ≤ c ≤ d ≤ e. Heinrich showed the existence of a partitioned incomplete Latin square (PILS) of type (a, b, c) and (a, b, c, d) if and only if a = b = c and 2a ≥ d. For PILS of type (a,b,c,d,e), it is necessary that a + b + c ≥ e, but not sufficient, and no characterization is currently known. In this talk we provide an additional necessary condition, classify the existence of PILS of type (a, b, c, d, a + b + c) and PILS with three equal parts, and show the existence of a family of PILS in which the parts are nearly the same size

A few remarks on avoiding partial Latin squares

Let P be an n x n array of symbols. P is called avoidable if for every set of n symbols, there is an n x n Latin square Lon these symbols so that corresponding cells in Land P differ. Due to recent work of Cavenagh and Olunan, we now know that all n x n partial Latin squares are avoidable for n 2: 4. Cavenagh and Ohman have shown that partial Latin squares of order 4m + 1 form 2: 1 [lJ and 4m -1 form 2: 2 [2) are avoidable. We give a short argument that includes all partial Latin squares of these orders of at least 9. We then ask the following question: given an n x n partial Latin square P with some specified structure, is there an n X n Latin square L of the same structure for which L avoids P? We answer this question in the context of generalized sudoku squares.Journal ArticlePublishe

On the existence of partitioned incomplete Latin squares with five parts

Let a, b, c, d, and e be positive integers. In 1982 Heinrich showed the existence of a partitioned incomplete Latin square (PILS) of type (a, b, c) and (a, b, c, d) if and only if a = b = c and 2a ≥ d. For PILS of type (a, b, c, d, e) with a ≤ b ≤ c ≤ d ≤ e, it is necessary that a+b+c ≥ e, but not sufficient. In this paper we prove an additional necessary condition and classify the existence of PILS of type (a, b, c, d, a + b + c) and PILS with three equal parts. Lastly, we show the existence of a family of PILS in which the parts are nearly the same size.Journal ArticlePublishe

On the chromatic index of Latin squares

A proper coloring of a Latin square of order n is an assignment of colors to its elements triples such that each row, column and symbol is assigned n distinct colors. Equivalently, a proper coloring of a Latin square is a partition into partial transversals. The chromatic index of a Latin square is the least number of colors needed for a proper coloring. We study the chromatic index of the cyclic Latin square which arises from the addition table for the integers modulo n. We obtain the best possible bounds except for the case when n=2 is odd and divisible by 3. We make some conjectures about the chromatic index, suggesting a generalization of Ryser's conjecture (that every Latin square of odd order contains a transversal).Journal ArticlePublishe

Hamilton cycle decompositions of k-uniform k-partite hypergraphs

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Some partial Latin cubes and their completions

It is well known that all n×n partial Latin squares with at most n−1 entries are completable. Our intent is to extend this well known statement to partial Latin cubes. We show that if an n×n×n partial Latin cube contains at most n − 1 entries, no two of which occupy the same row, then the partial Latin cube is completable. Also included in this paper is the problem of completing 2×n×n partial Latin boxes with at most n − 1 entries. Given certain sufficient conditions, we show when such partial Latin boxes are completable and then extendable to a deeper Latin box.Journal ArticlePublishe

Constrained completion of partial latin squares

In this paper, we combine the notions of completing and avoiding partial latin squares. Let P be a partial latin square of order n and let be the set of partial latin squares of order n that avoid P. We say that P is Q-completable if P can be completed to a latin square that avoids Q ∈ . We prove that if P has order 4t and contains at most t − 1 entries, then P is Q-completable for each Q ∈ when t ≥ 9.Journal ArticlePublishe