| Title:
|
Overlapping latin subsquares and full products (English) |
| Author:
|
Browning, Joshua M. |
| Author:
|
Vojtěchovský, Petr |
| Author:
|
Wanless, Ian M. |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
51 |
| Issue:
|
2 |
| Year:
|
2010 |
| Pages:
|
175-184 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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 $\frac nm{n\choose h}/{m\choose h}$ 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. (English) |
| Keyword:
|
latin square |
| Keyword:
|
latin subsquare |
| Keyword:
|
overlapping latin subsquares |
| Keyword:
|
full product in loops |
| MSC:
|
05B15 |
| MSC:
|
20N05 |
| idZBL:
|
Zbl 1224.05061 |
| idMR:
|
MR2682472 |
| . |
| Date available:
|
2010-05-21T12:41:39Z |
| Last updated:
|
2014-07-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/140097 |
| . |
| Reference:
|
[1] Dénes J., Hermann P.: On the product of all elements in a finite group.Ann. Discrete Math. 15 (1982), 105–109. MR 0772587 |
| Reference:
|
[2] Heinrich K., Wallis W.D.: The maximum number of intercalates in a latin square.Lecture Notes in Math. 884 (1981), 221–233. Zbl 0475.05014, MR 0641250, 10.1007/BFb0091822 |
| Reference:
|
[3] McKay B.D., Wanless I.M.: Most latin squares have many subsquares.J. Combin. Theory Ser. A 86 (1999), 323–347. Zbl 0948.05014, MR 1685535, 10.1006/jcta.1998.2947 |
| Reference:
|
[4] 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. MR 2505099 |
| Reference:
|
[5] Ryser H.J.: A combinatorial theorem with an application to latin rectangles.Proc. Amer. Math. Soc. 2 (1951), 550–552. Zbl 0043.01202, MR 0042361, 10.1090/S0002-9939-1951-0042361-0 |
| Reference:
|
[6] van Rees G.H.J.: Subsquares and transversals in latin squares.Ars Combin. 29B (1990), 193–204. Zbl 0718.05014, MR 1412875 |
| . |
