Complete Diagonals of Latin Squares

1979 ◽  
Vol 22 (4) ◽  
pp. 477-481 ◽  
Author(s):  
Gerard J. Chang
Keyword(s):  
Latin Square ◽  
Latin Squares ◽  
Special Case

AbstractJ. Marica and J. Schönhein [4], using a theorem of M. Hall, Jr. [3], see below, proved that if any n − 1 arbitrarily chosen elements of the diagonal of an n × n array are prescribed, it is possible to complete the array to form an n × n latin square. This result answers affirmatively a special case of a conjecture of T. Evans [2], to the effect that an n × n incomplete latin square with at most n − 1 places occupied can be completed to an n × n latin square. When the complete diagonal is prescribed, it is easy to see that a counterexample is provided by the case that one letter appears n − 1 times on the diagonal and a second letter appears once. In the present paper, we prove that except in this case the completion to a full latin square is always possible. Completion to a symmetric latin square is also discussed.

scholarly journals A Discussion of a Cryptographical Scheme Based in F-Critical Sets of a Latin Square

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 285
Author(s):  
Laura M. Johnson ◽  
Stephanie Perkins
Keyword(s):  
Necessary Conditions ◽  
Latin Square ◽  
Latin Squares ◽  
Critical Sets ◽  
Long Cycles ◽  
The Given

This communication provides a discussion of a scheme originally proposed by Falcón in a paper entitled “Latin squares associated to principal autotopisms of long cycles. Applications in cryptography”. Falcón outlines the protocol for a cryptographical scheme that uses the F-critical sets associated with a particular Latin square to generate access levels for participants of the scheme. Accompanying the scheme is an example, which applies the protocol to a particular Latin square of order six. Exploration of the example itself, revealed some interesting observations about both the structure of the Latin square itself and the autotopisms associated with the Latin square. These observations give rise to necessary conditions for the generation of the F-critical sets associated with certain autotopisms of the given Latin square. The communication culminates with a table which outlines the various access levels for the given Latin square in accordance with the scheme detailed by Falcón.

scholarly journals Atomic Latin Squares based on Cyclotomic Orthomorphisms

10.37236/1919 ◽  
2005 ◽  
Vol 12 (1) ◽  
Author(s):  
Ian M. Wanless
Keyword(s):  
Finite Fields ◽  
Complete Graph ◽  
Prime Order ◽  
Automorphism Groups ◽  
Latin Square ◽  
Latin Squares ◽  
Established Method ◽  
Cyclic Groups ◽  
Complete Bipartite Graphs ◽  
First Time

Atomic latin squares have indivisible structure which mimics that of the cyclic groups of prime order. They are related to perfect $1$-factorisations of complete bipartite graphs. Only one example of an atomic latin square of a composite order (namely 27) was previously known. We show that this one example can be generated by an established method of constructing latin squares using cyclotomic orthomorphisms in finite fields. The same method is used in this paper to construct atomic latin squares of composite orders 25, 49, 121, 125, 289, 361, 625, 841, 1369, 1849, 2809, 4489, 24649 and 39601. It is also used to construct many new atomic latin squares of prime order and perfect $1$-factorisations of the complete graph $K_{q+1}$ for many prime powers $q$. As a result, existence of such a factorisation is shown for the first time for $q$ in $\big\{$529, 2809, 4489, 6889, 11449, 11881, 15625, 22201, 24389, 24649, 26569, 29929, 32041, 38809, 44521, 50653, 51529, 52441, 63001, 72361, 76729, 78125, 79507, 103823, 148877, 161051, 205379, 226981, 300763, 357911, 371293, 493039, 571787$\big\}$. We show that latin squares built by the 'orthomorphism method' have large automorphism groups and we discuss conditions under which different orthomorphisms produce isomorphic latin squares. We also introduce an invariant called the train of a latin square, which proves to be useful for distinguishing non-isomorphic examples.

Pairs of Latin Squares to Counterbalance Sequential Effects and Pairing of Conditions and Stimuli

1989 ◽  
Vol 33 (18) ◽  
pp. 1223-1227 ◽  
Author(s):  
James R. Lewis
Keyword(s):  
Human Factors ◽  
Repeated Measures ◽  
Random Sequence ◽  
Latin Square ◽  
Latin Squares ◽  
Sequential Effects ◽  
Long Run ◽  
Within Subjects ◽  
Human Factors Research

This paper discusses methods with which one can simultaneously counterbalance immediate sequential effects and pairing of conditions and stimuli in a within-subjects design using pairs of Latin squares. Within-subjects (repeated measures) experiments are common in human factors research. The designer of such an experiment must develop a scheme to ensure that the conditions and stimuli are not confounded, or randomly order stimuli and conditions. While randomization ensures balance in the long run, it is possible that a specific random sequence may not be acceptable. An alternative to randomization is to use Latin squares. The usual Latin square design ensures that each condition appears an equal number of times in each column of the square. Latin squares have been described which have the effect of counterbalancing immediate sequential effects. The objective of this work was to extend these earlier efforts by developing procedures for designing pairs of Latin squares which ensure complete counterbalancing of immediate sequential effects for both conditions and stimuli, and also ensure that conditions and stimuli are paired in the squares an equal number of times.

A Partial Generalization of Mann's Theorem Concerning Orthogonal Latin Squares

1988 ◽  
Vol 31 (4) ◽  
pp. 409-413 ◽  
Author(s):  
E. T. Parker ◽  
Lawrence Somer
Keyword(s):  
Latin Square ◽  
Latin Squares ◽  
Necessary Condition ◽  
Orthogonal Latin Squares ◽  
Mutually Orthogonal Latin Squares ◽  
Complete Set

AbstractLetn = 4t+- 2, where the integert ≧ 2. A necessary condition is given for a particular Latin squareLof ordernto have a complete set ofn — 2mutually orthogonal Latin squares, each orthogonal toL.This condition extends constraints due to Mann concerning the existence of a Latin square orthogonal to a given Latin square.

Two remarks about Sudoku squares

2006 ◽  
Vol 90 (519) ◽  
pp. 425-430 ◽  
Author(s):  
A. D. Keedwell
Keyword(s):  
Current Problem ◽  
Latin Square ◽  
Latin Squares ◽  
Critical Set ◽  
Critical Sets ◽  
The Given ◽  
A Current ◽  
Defining Sets

Smallest defining setsA standard Sudoku square is a 9 × 9 latin square in which each of the nine 3 × 3 subsquares into which it can be separated contains each of the integers 1 to 9 exactly once.A current problem is to complete such a square when only some of the cells have been filled. These cells are often called ‘givens’. (Such problems are currently (2005) published daily in British newspapers.) In more mathematical terms, the given filled cells constitute a defining set or uniquely completable set for the square if they lead to a unique completion of the square. If, after deletion of any one of these givens, the square can no longer be completed uniquely, the givens form a critical set. The investigation of critical sets for ‘ordinary’ latin squares is a topic of current mathematical interest. (See [1] for more details.)

scholarly journals An Efficient Algorithm for Eigenvalue Problem of Latin Squares in a Bipartite Min-Max-Plus System

Symmetry ◽  
2020 ◽  
Vol 12 (2) ◽  
pp. 311 ◽  
Author(s):  
Mubasher Umer ◽  
Umar Hayat ◽  
Fazal Abbas ◽  
Anurag Agarwal ◽  
Petko Kitanov
Keyword(s):  
Eigenvalue Problem ◽  
Efficient Algorithm ◽  
Time Complexity ◽  
Latin Square ◽  
Latin Squares ◽  
Clear Advantage

In this paper, we consider the eigenproblems for Latin squares in a bipartite min-max-plus system. The focus is upon developing a new algorithm to compute the eigenvalue and eigenvectors (trivial and non-trivial) for Latin squares in a bipartite min-max-plus system. We illustrate the algorithm using some examples. The proposed algorithm is implemented in MATLAB, using max-plus algebra toolbox. Computationally speaking, our algorithm has a clear advantage over the power algorithm presented by Subiono and van der Woude. Because our algorithm takes 0 . 088783 sec to solve the eigenvalue problem for Latin square presented in Example 2, while the compared one takes 1 . 718662 sec for the same problem. Furthermore, a time complexity comparison is presented, which reveals that the proposed algorithm is less time consuming when compared with some of the existing algorithms.

scholarly journals Hownotto prove the Alon-Tarsi conjecture

2012 ◽  
Vol 205 ◽  
pp. 1-24 ◽  
Author(s):  
Douglas S. Stones ◽  
Ian M. Wanless
Keyword(s):  
Latin Square ◽  
Latin Squares ◽  
Exact Computation

AbstractThe sign of a Latin square is −1 if it has an odd number of rows and columns that are odd permutations; otherwise, it is +1. LetLEnandLonbe, respectively, the number of Latin squares of ordernwith sign +1 and −1. The Alon-Tarsi conjecture asserts thatLEn≠Lonwhennis even. Drisko showed thatLEp+1≢Lop+1(modp3) for primep≥ 3 and asked if similar congruences hold for orders of the formpk+ 1,p+ 3, orpq+ 1. In this article we show that ift≤n, thenLEn+1≢L0n+1(modt3) only ift = nandnis an odd prime, thereby showing that Drisko’s method cannot be extended to encompass any of the three suggested cases. We also extend exact computation ton≤ 9, discuss asymptotics forLo/LE, and propose a generalization of the Alon-Tarsi conjecture.

Circadian variation in the apparent digetibility of diets measured at the terminal ileum in pigs

1980 ◽  
Vol 94 (2) ◽  
pp. 399-405 ◽  
Author(s):  
R. M. Livingstone ◽  
Barbara A. Baird ◽  
T. Atkinson ◽  
R. M. J. Crofts
Keyword(s):  
Feed Intake ◽  
Terminal Ileum ◽  
Fish Meal ◽  
Circadian Variation ◽  
Latin Square ◽  
Latin Squares ◽  
Apparent Digestibility ◽  
Feeding Methods ◽  
Soyabean Meal ◽  
Ad Libitum

SummaryCircadian variation in the apparent digestibility of diets having different physical characteristics was measured in samples taken from the terminal ileum using simple cannulae and marker ratios. Six pigs were used in three latin squares involving three iso-nitrogenous diets (30 g N/kg D.M.). Diet A was based on barley, weatings, soyabean meal and fish meal, diet B included barley, weatings and oats and diet C was purified. The allocation of diet provided 100 g D.M./kg Weg0·75/24 h and in each latin square a different pattern of feed intake was used; diets were given at intervals of either 1 or 12 h, or ad libitum.Differences in the digestibility of the diets were consistently distinguished by the technique. The circadian variation in digestibility was related to the type of diet and could be modified by changing the number and distribution of feeds per day. The results show that an understanding of the variation associated with different diets and feeding methods is necessary for optimizing the strategy for sampling from the terminal ileum.

MAXIMUM GENUS EMBEDDINGS OF LATIN SQUARES

2017 ◽  
Vol 60 (2) ◽  
pp. 495-504 ◽  
Author(s):  
TERRY S. GRIGGS ◽  
CONSTANTINOS PSOMAS ◽  
JOZEF ŠIRÁŇ
Keyword(s):  
Latin Square ◽  
Latin Squares ◽  
Orientable Surface ◽  
Maximum Genus ◽  
Odd Order

AbstractIt is proved that every non-trivial Latin square has an upper embedding in a non-orientable surface and every Latin square of odd order has an upper embedding in an orientable surface. In the latter case, detailed results about the possible automorphisms and their actions are also obtained.

scholarly journals On the completion of latin rectangles to symmetric latin squares

2004 ◽  
Vol 76 (1) ◽  
pp. 109-124 ◽  
Author(s):  
Darryn Bryant ◽  
C. A. Rodger
Keyword(s):  
Sufficient Conditions ◽  
Latin Square ◽  
Latin Squares ◽  
Necessary And Sufficient Conditions ◽  
Latin Rectangle ◽  
Necessary And Sufficient ◽  
Edge Colouring

AbstractWe find necessary and sufficient conditions for completing an arbitrary 2 by n latin rectangle to an n by n symmetric latin square, for completing an arbitrary 2 by n latin rectangle to an n by n unipotent symmetric latin square, and for completing an arbitrary 1 by n latin rectangle to an n by n idempotent symmetric latin square. Equivalently, we prove necessary and sufficient conditions for the existence of an (n−1)-edge colouring of Kn (n even), and for n-edge colouring of Kn (n odd) in which the colours assigned to the edges incident with two vertices are specified in advance.

Sign in / Sign up

Export Citation Format

Share Document