On the Existence of Partitioned Incomplete Latin Squares with Five Parts

Marshall University Marshall Digital Scholar Mathematics Faculty Research Mathematics 2019 On the Existence of Partitioned Incomplete Latin Squares with Five Parts Jaromy Kuhl Donald McGinn Michael W. Schroeder Follow this and additional works at: https://mds.marshall.edu/mathematics_faculty Part of the Discrete Mathematics and Combinatorics Commons AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 74(1) (2019), Pages 46–60 On the existence of partitioned incomplete Latin squares with five parts Jaromy Kuhl Donald McGinn Department of Mathematics and Statistics University of West Florida Pensacola, FL 32514 U.S.A. [email protected] [email protected] Michael William Schroeder Department of Mathematics Marshall University Huntington, WV 25755 U.S.A. [email protected] Abstract Let a, b, c, d,ande 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)andPILS with three equal parts. Lastly, we show the existence of a family of PILS in which the parts are nearly the same size. 1 Introduction Let a, n ∈ Z+ and S be a symbol set of order n.Let[n]={1,...,n}, a+S = {a+s | s ∈ S},andaS be the multiset in which each element of S occurs a times. A partial Latin square of order n over S is a partially filled n×n matrix in which symbols from S occur at most once in each row and column. Unless otherwise stated, we assume that S =[n]. We use PLS(n) to denote the set of partial Latin squares of order n. Let P ∈ PLS(n). The content of cell (i, j)ofP for each i, j ∈ [n] is denoted P (i, j) and either P (i, j) ∈ S or P (i, j)=∅.IfP (i, j) is nonempty for each i, j ∈ [n], then ISSN: 2202-3518 c The author(s). Released under the CC BY-ND 4.0 International License J. KUHL ET AL. / AUSTRALAS. J. COMBIN. 74 (1) (2019), 46–60 47 P is called a Latin square and P ∈ LS(n), where LS(n) is the set of Latin squares of order n. We will often treat P as a subset of [n] × [n] × [n], where (i, j, k) ∈ P if and only if k = P (i, j). We say that P can be completed over [n] if there exists L ∈ LS(n)over[n] containing P . There are numerous results in the literature on completing partial Latin squares. Let Sn be the symmetric group acting on [n]. For θ =(α, β, γ) ∈ Sn × Sn × Sn,we use θ(P ) ∈ PLS(n) to denote the matrix in which the rows, columns, and symbols of P are permuted according to α, β,andγ respectively. The mapping θ is called an isotopism,andP and θ(P ) are said to be isotopic. It is well known, and follows from a straightforward argument, that for any θ ∈ Sn × Sn × Sn, θ(P ) ∈ PLS(n)and P can be completed if and only if θ(P )canbecompleted. In what follows we review two collections of objects: partitioned incomplete Latin squares (PILS) and Latin squares realizing partitions. The former is well-documented (see III.1.5 in [3]) and we establish that the latter objects are equivalent. Let H = {S1,S2, ..., Sm} be a partition of [n], where |Si| = si for each i ∈ [m]. A partitioned incomplete Latin square (PILS) of order n, having partition H,isa partial Latin square P of order n indexed by [n] satisfying the following properties: (1) The cells Si × Si are empty for each i ∈ [m]. (2) For each j ∈ Si,rowj and column j of P contains every element of [n]\Si exactly once. If P is a PILS of order n with partition H,thenθ(P ) is also a PILS of order n for each θ ∈ Sn ×Sn ×Sn.Itmaybethatθ(P ) is under a new partition H = {S1,S2, ..., Sm}. However, it holds that |Si| = si for each i ∈ [m]. Thus, we may assume that Si =(s1 + s2 + ...+ si−1)+[si]foreachi ∈ [m]. The set of all PILS of order n with partition H is denoted as PILS(s1,...,sm). We also say that an element of PILS(s1,...,sm)isaPILSoftype(s1,...,sm). The purpose of this paper is to find necessary and sufficient conditions under which PILS(s1,...,sm) is nonempty when m =5.(Thereasonwefocusonm = 5 is because for m ≤ 4, necessary and sufficient conditions are known; see Theorem 1.1 below.) 5 4 2 3 1 5 4 2 3 4 5 3 1 4 2 5 3 1 5 4 1 2 5 4 3 1 2 2 3 1 2 3 1 4 5 3 1 2 3 1 2 5 4 Figure 1: A PILS of type (1, 1, 1, 2) and the corresponding LS realizing (13, 2). Let L ∈ LS(n)andq<nbe a positive integer. If M is a q ×q sub-matrix of L and is itself a Latin square, then M is called a subsquare of L. Subsquares of L are disjoint if they do not share a row, column, or symbol. In Figure 1, four pairwise disjoint subsquares are highlighted in the Latin square. Let (n1,...,nk) be an integer partition of n. We may abbreviate this if parts of the α1 α partition are identical; if n = α1n1 + ...+ αn then we may also let (n1 ,...,n ) J. KUHL ET AL. / AUSTRALAS. J. COMBIN. 74 (1) (2019), 46–60 48 denote the partition of n with αi copies of ni for each i ∈ []. We say that (n1,...,nk) is realized if there is a Latin square of order n with pairwise disjoint subsquares of order ni for each i ∈ [k]. For example, the Latin square in Figure 1 realizes 3 (1 , 2). Observe that P ∈ PILS(s1, ..., sm) if and only if the completion of P is a realization of (s1, ..., sm). Hence there is a one-to-one correspondence between PILS and Latin squares that realize integer partitions. Many of the results stated below were originally stated as realizations of integer partitions. We will state them here as existence results of PILS. It is straightforward that PILS(a, n − a)isemptyforeacha ∈ [n]. Heinrich [6] classified the existence of PILS having partitions with exactly 3 or 4 parts. Theorem 1.1. Let a, b, c,andd be positive integers. 1. PILS(a, b, c) is nonempty if and only if a = b = c. 2. PILS(a, b, c, d) is nonempty if and only if a = b = c and 1 ≤ d ≤ 2a (perhaps after relabeling). 3. PILS(ak) is nonempty if and only if k =2 . In [7], Heinrich constructed Latin squares realizing partitions with exactly two dis- tinct parts and proved statements 1 and 2 in Theorem 1.2 below. In [10], Kuhl and Schroeder proved the third statement in Theorem 1.2, which together with Theorem 1.1 completely characterizes PILS of type (as,bt). Theorem 1.2. Let a and b be any positive integers. 1. If s ≥ 3 and t ≥ 3,thenPILS(as,bt) is nonempty. 2. PILS(as,b) is nonempty if and only if (s − 1)a ≥ b and s ≥ 3. 3. PILS(as,b2) is nonempty if and only if sa ≥ b and s ≥ 3. A necessary condition for a PILS of type having k parts is that each part is bounded by the sum of k − 2 other parts, but it is not sufficient. In particular, it is known that PILS of type (n1,n2,n3,n3,n1 + n2 + n3) do not exist [9], and in [9] it is stated that there are numerous examples of PILS of five parts that do not exist. This leads us to believe that characterizing the existence of these PILS is difficult. In Section 2 we define outline squares and illustrate their relevance to finding PILS. In Section 3 we give a new necessary condition for PILS of five parts to exist. In Section 4 we use outline squares to prove necessary and sufficient conditions under which PILS of type (n1,n2,n3,n4,n1 + n2 + n3) and PILS with three equal parts exist. In Section 5, we use an inductive technique to show the existence of PILS where the parts are close in size. Finally, we end with a conjecture and a discussion on how the existence problem can be translated into a integer programming problem. We also give some comparisons between the results of this paper to some by Colbourn in a recent publication [2]. J. KUHL ET AL. / AUSTRALAS. J. COMBIN. 74 (1) (2019), 46–60 49 2 Outline Squares We begin with arrays in which cells contain more than one symbol and symbols may occur more than once in a cell, row, and column. The following constructions are attributed to Hilton [8], though his constructions are more general. Definitions and notations come from [8]. Let Q =(q1,...,qu) be a partition of n.LetL ∈ LS(n). The reduction modulo (Q, Q, Q)ofL is made by amalgamating rows (q1 +···+qi−1)+[qi], columns (q1 +···+qi−1)+[qi], and symbols (q1 +···+qi−1)+[qi]foreachi ∈ [u]. Thus, the reduction modulo (Q, Q, Q)isau×u array whose cells are filled from [u] such that symbol k occurs t times in cell (i, j) if and only if symbols from (q1 +···+qk−1)+[qk] occur exactly t times in the corresponding sub-matrix of L.

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