Partial Latin Research Articles

Toward a three-dimensional counterpart of Cruse’s theorem

Completing partial latin squares is NP-complete. Motivated by Ryser’s theorem for latin rectangles, in 1974, Cruse found conditions that ensure a partial symmetric latin square of order m m can be embedded in a symmetric latin square of order n n . Loosely speaking, this results asserts that an n n -coloring of the edges of the complete m m -vertex graph K m K_m can be embedded in a one-factorization of K n K_n if and only if n n is even and the number of edges of each color is at least m − n / 2 m-n/2 . We establish necessary and sufficient conditions under which an edge-coloring of the complete λ \lambda -fold m m -vertex 3-graph λ K m 3 \lambda K_m^3 can be embedded in a one-factorization of λ K n 3 \lambda K_n^3 . In particular, we prove the first known Ryser type theorem for hypergraphs by showing that if n ≡ 0 ( m o d 3 ) n\equiv 0\ (\mathrm {mod}\ 3) , any edge-coloring of λ K m 3 \lambda K_m^3 where the number of triples of each color is at least m / 2 − n / 6 m/2-n/6 , can be embedded in a one-factorization of λ K n 3 \lambda K_n^3 . Finally we prove an Evans type result by showing that if n ≡ 0 ( m o d 3 ) n\equiv 0\ (\mathrm {mod}\ 3) and n ≥ 3 m n\geq 3m , then any q q -coloring of the edges of any F ⊆ λ K m 3 F\subseteq \lambda K_m^3 can be embedded in a one-factorization of λ K n 3 \lambda K_n^3 as long as q ≤ λ ( n − 1 2 ) − λ ( m 3 ) / ⌊ m / 3 ⌋ q\leq \lambda \binom {n-1}{2}-\lambda \binom {m}{3}/\left \lfloor m/3\right \rfloor .

On maximal partial Latin hypercubes

A lower bound is presented for the minimal number of filled cells in a maximal partial Latin hypercube of dimension d and order n. The result generalises and extends previous results for d=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d=2$$\\end{document} (Latin squares) and d=3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d=3$$\\end{document} (Latin cubes). Explicit constructions show that this bound is near-optimal for large n>d\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n> d$$\\end{document}. For d>n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d>n$$\\end{document}, a connection with Hamming codes shows that this lower bound gives a related upper bound for the same quantity. The results can be interpreted in terms of independent dominating sets in certain graphs, and in terms of codes that have covering radius 1 and minimum distance at least 2.

A massively parallel evolutionary algorithm for the partial Latin square extension problem

The partial Latin square extension problem is to fill as many as possible empty cells of a partially filled Latin square. This problem is a useful model for a wide range of applications in diverse domains. This paper presents the first massively parallel evolutionary algorithm for this computationally challenging problem based on a transformation of the problem to partial graph coloring. The algorithm features the following original elements. Based on a very large population (with more than 104 individuals) and modern graphical processing units, the algorithm performs many local searches in parallel to ensure an intensive exploitation of the search space. The algorithm employs a dedicated crossover with a specific parent matching strategy to create a large number of diversified and information-preserving offspring at each generation. Extensive experiments on 1800 benchmark instances show a high competitiveness of the algorithm compared to the current best performing methods. Competitive results are also reported on the related Latin square completion problem. Analyses are performed to shed lights on the roles of the main algorithmic components.

Completing partial Latin squares with two filled rows and three filled columns

Consider a partial Latin square $P$ where the first two rows and first three columns are completely filled, and every other cell of $P$ is empty. It has been conjectured that all such partial Latin squares of order at least $8$ are completable. Based on a technique by Kuhl and McGinn we describe a framework for completing partial Latin squares in this class. Moreover, we use our method for proving that all partial Latin squares from this family, where the intersection of the nonempty rows and columns form a Latin rectangle with three distinct symbols, is completable.

Effects of supplementing a yeast culture in a pelleted total mixed ration on fiber degradation, fermentation parameters, and the bacterial community in the rumen of sheep

Feeding animals with pelleted total mixed ration (TMR) is a widespread practice as it reduces feed losses during storage, feeding and the animal’s sorting. Supplementation of yeast culture can promote rumen fiber degradation, which may occur through stabilizing rumen pH. There are few studies, however, evaluating the optimal supplementation level of yeast culture in pelleted TMR in which rumen pH has been continuously measured. This study aimed to determine the optimal level of a commercial yeast culture product (of which the number of viable CFU was not measured) mixed and pelleted with the TMR, on fiber degradation in situ, rumen pH dynamics, rumen fermentation variables, and the bacterial community composition of rumen fluid. Under a partial Latin square design, a total of twelve wethers equipped with permanent rumen fistulas were randomly assigned to four treatments of yeast culture at 0, 0.5, 1.0, or 2.0% dietary dry matter. After a 28 d period comprising 21 d for adaptation and 7 d for measurements, each group of three wethers was randomly reassigned to another treatment. Thus, each yeast supplementation level was fed to a total of six animals. In situ dry matter (DMD) and neutral detergent fiber (NDFD) disappearance of corn stover, mean rumen pH, total volatile fatty acids concentration, acetate, propionate, and n-butyrate molar percentages, and the acetate to propionate ratio were influenced (P ≤ 0.05) by dietary treatments. Quadratic (P ≤ 0.03) responses to increasing yeast culture supplementation level were observed for mean rumen pH, daily duration of pH below a 5.8 threshold considered inhibitory for fibrolysis, and the molar percentages of propionate, n-butyrate, and the acetate to propionate ratio. The highest in situ DMD and NDFD occurred when yeast was supplemented at 1%. Increasing yeast culture supplementation level tended to quadratically increase the abundance of fibrolytic bacterial genera NK4A214 and FD2005, which peaked at 1% yeast culture in the diet DM. The molar percentage of n-butyrate was positively correlated with the relative abundance of Rikenellaceae_RC9 and NK4A214. In our study, supplementation of pelleted TMR with 1% yeast culture provided as residual yeast medium components, was optimal for fiber degradation of forage fed to ruminants. Members of genera NK4A214 and FD2005 may contribute extra n-butyrate.

On Completing Partial Latin Squares with Prescribed Diagonals

Necessary and sufficient numerical conditions are known for the embedding of an incomplete latin square $L$ of order $n$ into a latin square $T$ of order $t \geq 2n+1$ in which each symbol is prescribed to occur in a given number of cells on the diagonal of $T$ outside of $L$. This includes the classic case where $T$ is required to be idempotent. If $t<2n$ then no such numerical sufficient conditions exist since it is known that the arrangement of symbols within the given incomplete latin square can determine the embeddibility. All examples where the arrangement is a factor share the common feature that one symbol is prescribed to appear exactly once in the diagonal of $T$ outside of $L$, resulting in a conjecture over 30 years ago stating that it is only this feature that prevents numerical conditions sufficing for all $t \geq n$.In this paper we prove this conjecture, providing necessary and sufficient numerical conditions for the embedding of an incomplete latin square $L$ of order $n$ into a latin square $T$ of order $t$ for all $t \geq n$ in which the diagonal of $T$ outside of $L$ is prescribed in the case where no symbol is required to appear exactly once in the diagonal of $T$ outside of $L$.

An Evans-Style Result for Block Designs

For positive integers $n$ and $k$ with $n \geq k$, an $(n,k,1)$-design is a pair $(V, \mathcal{B})$ where $V$ is a set of $n$ points and $\mathcal{B}$ is a collection of $k$-subsets of $V$ called blocks such that each pair of points occur together in exactly one block. If we weaken this condition to demand only that each pair of points occur together in at most one block, then the resulting object is a partial $(n,k,1)$-design. A completion of a partial $(n,k,1)$-design $(V,\mathcal{A})$ is a (complete) $(n,k,1)$-design $(V,\mathcal{B})$ such that $\mathcal{A} \subseteq \mathcal{B}$. Here, for all sufficiently large $n$, we determine exactly the minimum number of blocks in an uncompletable partial $(n,k,1)$-design. This result is reminiscent of Evans' now-proved conjecture on completions of partial latin squares. We also prove some related results concerning edge decompositions of almost complete graphs into copies of $K_k$.

The Four Signs of the Art: Edition and Translation of an Alchemical Epistle Attributed to Ḫālid b. Yazīd and its Latin Translation

Abstract In the article, the authors present a study, a critical edition and an English translation of an Arabic alchemical epistle attributed to the Umayyad prince Ḫālid b. Yazīd, together with its Latin translation recently identified by the authors. Among the many alchemical works attributed to Ḫālid b. Yazīd, this untitled Risāla (inc.: ‮إني رأيت الناس طلبوا صنعة الحكمة‬‎) is the second most represented in the manuscript tradition, with no less than twelve witnesses containing it. Its partial Latin translation, available in six manuscripts, was also attributed to Calid, but the name of the translator remains unknown.

Recognition and Analysis of Image Patterns Based on Latin Squares by Means of Computational Algebraic Geometry

With the particular interest of being implemented in cryptography, the recognition and analysis of image patterns based on Latin squares has recently arisen as an efficient new approach for classifying partial Latin squares into isomorphism classes. This paper shows how the use of a Computer Algebra System (CAS) becomes necessary to delve into this aspect. Thus, the recognition and analysis of image patterns based on these combinatorial structures benefits from the use of computational algebraic geometry to determine whether two given partial Latin squares describe the same affine algebraic set. This paper delves into this topic by focusing on the use of a CAS to characterize when two partial Latin squares are either partial transpose or partial isotopic.

Constructing and embedding mutually orthogonal Latin squares: reviewing both new and existing results

We review results for the embedding of orthogonal partial Latin squares in orthogonal Latin squares, comparing and contrasting these with results for embedding partial Latin squares in Latin squares. We also present a new construction that uses the existence of a set of $t$ mutually orthogonal Latin squares of order $n$ to construct a set of $2t$ mutually orthogonal Latin squares of order $n^t$.

Latin squares from multiplication tables

AbstractConsider the usual multiplication table in which all entries strictly greater than were deleted. Can we fill in the empty places so that the resulting table will become a multiplication table of a group? This intriguing question arose in connection to the famous Graham's Greatest Common Divisor Problem. We prove here a weaker statement, namely, that one may always complete such a partial multiplication table to a Latin square. Our result is more general, as it guarantees the possibility of completion of any partial Latin square, provided that the shape of the empty part satisfies certain conditions. We also point on connections of our result to a far‐reaching extension of Graham's problem involving graphs and homogeneous arithmetic progressions.

Computing Autotopism Groups of Partial Latin Rectangles

Computing the autotopism group of a partial Latin rectangle (PLR) can be performed in multiple ways. This study has two aims: comparing some of these methods experimentally to identify those that are competitive; and identifying design goals for developing practical software. We compare six families of algorithms (two backtracking and four graph-theoretic methods), with and without using entry invariants (EIs), in a range of settings. Two EIs are considered: frequencies of row, column, and symbol representatives; and 2 × 2 submatrices. The best approach to computing autotopism groups varies. When PLRs have many autotopisms (such as having very few entries or being a group table), the McKay, Meynert, and Myrvold (MMM) method computes generators for the autotopism group efficiently. (The MMM method is the standard way to compute autotopisms.) Otherwise, PLRs ordinarily have trivial or small autotopism groups, and the task is to verify this. The so-called PLR graph method is slightly more efficient in this setting than the MMM method (in some circumstances, around twice as fast). With an intermediate number of entries, the quick-to-compute strong EIs are effective at reducing the need for computation without introducing significant overhead. With a full or almost-full PLR, a more sophisticated EI is needed to reduce down-the-line computation. These results suggest a hybrid approach to computing autotopism groups: The software decides on suitable EIs based on the input; and the user chooses between the MMM or the PLR graph methods, depending on their dataset. This article expands the authors’ previous article Computing autotopism groups of PLRs: a pilot study .

Enumerating Partial Latin Rectangles

This paper deals with different computational methods to enumerate the set $\mathrm{PLR}(r,s,n;m)$ of $r \times s$ partial Latin rectangles on $n$ symbols with $m$ non-empty cells. For fixed $r$, $s$, and $n$, we prove that the size of this set is given by a symmetric polynomial of degree $3m$, and we determine the leading terms (the monomials of degree $3m$ through $3m-9$) using inclusion-exclusion. For $m \leqslant 13$, exact formulas for these symmetric polynomials are determined using a chromatic polynomial method. Adapting Sade's method for enumerating Latin squares, we compute the exact size of $\mathrm{PLR}(r,s,n;m)$, for all $r \leqslant s \leqslant n \leqslant 7$, and all $r \leqslant s \leqslant 6$ when $n=8$. Using an algebraic geometry method together with Burnside's Lemma, we enumerate isomorphism, isotopism, and main classes when $r \leqslant s \leqslant n \leqslant 6$. Numerical results have been cross-checked where possible.

K-Plex 2-Erasure Codes and Blackburn Partial Latin Squares

A k-plex of order n is an n × n matrix on n symbols, where every row contains k distinct symbols, every column contains k distinct symbols, and every symbol occurs exactly k times. Yi et al. (2019) introduced 3-plex codes which are 2-erasure codes (2-erasure tolerant array codes) derived from 3-plexes. In this paper, we generalize 3-plex codes to k-plex codes. We introduce the notion of a “strong” k-plex which implies the derived k-plex code is 2-erasure tolerant. Moreover, k-plex codes derived from strong k-plexes have a straightforward algorithm for reconstruction. These general k-plex codes offer greater flexibility when choosing a suitable code for a storage system, enabling the operator to better optimize the unavoidable trade-offs involved. Blackburn asked for the maximum number of entries in an n × n partial Latin square on n symbols in which if distinct cells (i, j) and (i', j') contain the same symbol, then the cells (i', j) and (i, j') are empty. A “strong” k-plex satisfies the Blackburn property (along with two other properties related to erasure coding). We investigate the necessary conditions for the existence of Blackburn k-plexes (and hence necessary conditions for the existence of strong k-plexes). We show that any Blackburn k-plex has order n ≥ [(√2 + 1)k - 2]. We describe how to construct strong k-plexes of order n when k ∈ {2, 3, 4, 5} for all possible orders n, and we give a simple construction of strong k-plexes of order k <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> for k ≥ 2.

Computing autotopism groups of partial Latin rectangles: A pilot study

Computing the autotopism group of a partial Latin rectangle can be performed in a variety of ways. This pilot study has two aims: (a) to compare these methods experimentally, and (b) to identify the design goals one should have in mind for developing practical software. To this end, we compare six families of algorithms (two backtracking methods and four graph automorphism methods), with and without the use of entry invariants, on two test suites. We consider two entry invariants: one determined by the frequencies of row, column, and symbol representatives, and one determined by $2 \times 2$ submatrices. We find: (a) with very few entries, many symmetries often exist, and these should be identified mathematically rather than computationally, (b) with an intermediate number of entries, a quick-to-compute entry invariant was effective at reducing the need for computation, (c) with an almost-full partial Latin rectangle, more sophisticated entry invariants are needed, and (d) the performance for (full) Latin squares is significantly poorer than other partial Latin rectangles of comparable size, obstructed by the existence of Latin squares with large (possibly transitive) autotopism groups.

Edge precoloring extension of hypercubes

AbstractWe consider the problem of extending partial edge colorings of hypercubes. In particular, we obtain an analogue of the positive solution to the famous Evans' conjecture on completing partial Latin squares by proving that every proper partial edge coloring of at most edges of the ‐dimensional hypercube can be extended to a proper ‐edge coloring of . Additionally, we characterize which partial edge colorings of with precisely precolored edges are extendable to proper ‐edge colorings of .

New bounds for the Moser‐Tardos distribution

The Lovász local lemma (LLL) is a probabilistic tool to generate combinatorial structures with good “local” properties. The “LLL‐distribution” further shows that these structures have good global properties in expectation. The seminal algorithm of Moser and Tardos turned the simplest, variable‐based form of the LLL into an efficient algorithm; this has since been extended to other probability spaces including random permutations. One can similarly define an “MT‐distribution” for these algorithms, that is, the distribution of the configuration they produce. We show new bounds on the MT‐distribution in the variable and permutation settings which are significantly stronger than those known to hold for the LLL‐distribution. As some example illustrations, we show a nearly tight bound on the minimum implicate size of a CNF Boolean formula, and we obtain improved bounds on weighted Latin transversals and partial Latin transversals.

Embedding partial Latin squares in Latin squares with many mutually orthogonal mates

In this paper it is shown that any partial Latin square of order n can be embedded in a Latin square of order at most 16n2 which has at least 2n mutually orthogonal mates. Further, for any t⩾2, it is shown that a pair of orthogonal partial Latin squares of order n can be embedded in a set of t mutually orthogonal Latin squares (MOLS) of order a polynomial with respect to n. A consequence of the constructions is that, if N(n) denotes the size of the largest set of MOLS of order n, then N(n2)⩾N(n)+2. In particular, it follows that N(576)⩾9, improving the previously known lower bound N(576)⩾8.

Refining invariants for computing autotopism groups of partial Latin rectangles

Prior to using computational tools that find the autotopism group of a partial Latin rectangle (its stabilizer group under row, column and symbol permutations), it is beneficial to find partitions of the rows, columns and symbols that are invariant under autotopisms and are as fine as possible. We look at the lattices formed by these partitions and introduce two invariant refining maps on these lattices. The first map generalizes the strong entry invariant in a previous work. The second map utilizes some bipartite graphs, introduced here, whose structure is determined by pairs of rows (or columns, or symbols). Experimental results indicate that in most cases (ordinarily 99%+), the combined use of these invariants gives the theoretical best partition of the rows, columns and symbols, outperforms the strong entry invariant, which only gives the theoretical best partitions in roughly 80% of the cases.

Using a CAS/DGS to Analyze Computationally the Configuration of Planar Bar Linkage Mechanisms Based on Partial Latin Squares

Currently, the study of new isotopism invariants of partial Latin squares constitutes an open and active problem. This paper delves into this topic by analyzing computationally the configuration of planar bar linkage mechanisms for which the array formed by the lengths of their bars constitutes an empty-diagonal symmetric partial Latin square such that each one of its rows and columns has at least two non-empty cells. These assumptions enable one to define a series of algebraic and geometric constraints that can be readily implemented in any Computer Algebra or Dynamic Geometry System. In order to illustrate the different concepts and results introduced throughout the paper, it is explicitly determined and characterized the distribution of planar bar linkage mechanisms based on partial Latin squares of order up to five.