Abstract
Fuchs asked: For which integer partitions g1 +· · · gn = N does there exist a latin square of side N having n subsquares of sides g1, …,gn having no rows, columns, or symbols in common? Only when at most two distinct integer parts are used is the answer known completely; in general even the necessary conditions are elusive. Two conjectures giving plausible sufficient conditions are advanced. The first asserts than whenever the largest three integer parts are the same, such a latin square always exists. The second asserts that whenever the largest part is no larger than n − 2 times the smallest, such a latin square always exists. Partial results on both are established.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 501-536 |
| Number of pages | 36 |
| Journal | Australasian Journal of Combinatorics |
| Volume | 71 |
| Issue number | 3 |
| State | Published - Jan 1 2018 |
Fingerprint
ASJC Scopus subject areas
- Discrete Mathematics and Combinatorics
Cite this
On a latin square problem of fuchs. / Colbourn, Charles.
In: Australasian Journal of Combinatorics, Vol. 71, No. 3, 01.01.2018, p. 501-536.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - On a latin square problem of fuchs
AU - Colbourn, Charles
PY - 2018/1/1
Y1 - 2018/1/1
N2 - Fuchs asked: For which integer partitions g1 +· · · gn = N does there exist a latin square of side N having n subsquares of sides g1, …,gn having no rows, columns, or symbols in common? Only when at most two distinct integer parts are used is the answer known completely; in general even the necessary conditions are elusive. Two conjectures giving plausible sufficient conditions are advanced. The first asserts than whenever the largest three integer parts are the same, such a latin square always exists. The second asserts that whenever the largest part is no larger than n − 2 times the smallest, such a latin square always exists. Partial results on both are established.
AB - Fuchs asked: For which integer partitions g1 +· · · gn = N does there exist a latin square of side N having n subsquares of sides g1, …,gn having no rows, columns, or symbols in common? Only when at most two distinct integer parts are used is the answer known completely; in general even the necessary conditions are elusive. Two conjectures giving plausible sufficient conditions are advanced. The first asserts than whenever the largest three integer parts are the same, such a latin square always exists. The second asserts that whenever the largest part is no larger than n − 2 times the smallest, such a latin square always exists. Partial results on both are established.
UR - http://www.scopus.com/inward/record.url?scp=85046746189&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85046746189&partnerID=8YFLogxK
M3 - Article
VL - 71
SP - 501
EP - 536
JO - Australasian Journal of Combinatorics
JF - Australasian Journal of Combinatorics
SN - 1034-4942
IS - 3
ER -