# Overlapping latin subsquares and full products

Joshua M. Browning; Petr Vojtěchovský; Ian M. Wanless

Commentationes Mathematicae Universitatis Carolinae (2010)

- Volume: 51, Issue: 2, page 175-184
- ISSN: 0010-2628

## Access Full Article

top## Abstract

top## How to cite

topBrowning, Joshua M., Vojtěchovský, Petr, and Wanless, Ian M.. "Overlapping latin subsquares and full products." Commentationes Mathematicae Universitatis Carolinae 51.2 (2010): 175-184. <http://eudml.org/doc/37750>.

@article{Browning2010,

abstract = {We derive necessary and sufficient conditions for there to exist a latin square of order $n$ containing two subsquares of order $a$ and $b$ that intersect in a subsquare of order $c$. We also solve the case of two disjoint subsquares. We use these results to show that:
(a) A latin square of order $n$ cannot have more than $\frac\{n\}\{m\}\{n\atopwithdelims ()h\}/\{m\atopwithdelims ()h\}$ subsquares of order $m$, where $h=\lceil (m+1)/2\rceil $. Indeed, the number of subsquares of order $m$ is bounded by a polynomial of degree at most $\sqrt\{2m\}+2$ in $n$.
(b) For all $n\ge 5$ there exists a loop of order $n$ in which every element can be obtained as a product of all $n$ elements in some order and with some bracketing.},

author = {Browning, Joshua M., Vojtěchovský, Petr, Wanless, Ian M.},

journal = {Commentationes Mathematicae Universitatis Carolinae},

keywords = {latin square; latin subsquare; overlapping latin subsquares; full product in loops; Latin square; Latin subsquare; overlapping Latin subsquare; full product in loops},

language = {eng},

number = {2},

pages = {175-184},

publisher = {Charles University in Prague, Faculty of Mathematics and Physics},

title = {Overlapping latin subsquares and full products},

url = {http://eudml.org/doc/37750},

volume = {51},

year = {2010},

}

TY - JOUR

AU - Browning, Joshua M.

AU - Vojtěchovský, Petr

AU - Wanless, Ian M.

TI - Overlapping latin subsquares and full products

JO - Commentationes Mathematicae Universitatis Carolinae

PY - 2010

PB - Charles University in Prague, Faculty of Mathematics and Physics

VL - 51

IS - 2

SP - 175

EP - 184

AB - We derive necessary and sufficient conditions for there to exist a latin square of order $n$ containing two subsquares of order $a$ and $b$ that intersect in a subsquare of order $c$. We also solve the case of two disjoint subsquares. We use these results to show that:
(a) A latin square of order $n$ cannot have more than $\frac{n}{m}{n\atopwithdelims ()h}/{m\atopwithdelims ()h}$ subsquares of order $m$, where $h=\lceil (m+1)/2\rceil $. Indeed, the number of subsquares of order $m$ is bounded by a polynomial of degree at most $\sqrt{2m}+2$ in $n$.
(b) For all $n\ge 5$ there exists a loop of order $n$ in which every element can be obtained as a product of all $n$ elements in some order and with some bracketing.

LA - eng

KW - latin square; latin subsquare; overlapping latin subsquares; full product in loops; Latin square; Latin subsquare; overlapping Latin subsquare; full product in loops

UR - http://eudml.org/doc/37750

ER -

## References

top- Dénes J., Hermann P., On the product of all elements in a finite group, Ann. Discrete Math. 15 (1982), 105–109. MR0772587
- Heinrich K., Wallis W.D., 10.1007/BFb0091822, Lecture Notes in Math. 884 (1981), 221–233. Zbl0475.05014MR0641250DOI10.1007/BFb0091822
- McKay B.D., Wanless I.M., 10.1006/jcta.1998.2947, J. Combin. Theory Ser. A 86 (1999), 323–347. Zbl0948.05014MR1685535DOI10.1006/jcta.1998.2947
- Pula K., Products of all elements in a loop and a framework for non-associative analogues of the Hall-Paige conjecture, Electron. J. Combin. 16 (2009), R57. MR2505099
- Ryser H.J., 10.1090/S0002-9939-1951-0042361-0, Proc. Amer. Math. Soc. 2 (1951), 550–552. Zbl0043.01202MR0042361DOI10.1090/S0002-9939-1951-0042361-0
- van Rees G.H.J., Subsquares and transversals in latin squares, Ars Combin. 29B (1990), 193–204. Zbl0718.05014MR1412875

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.