### Abstract

Fuchs asked: For which integer partitions g_{1} +· · · g_{n} = N does there exist a latin square of side N having n subsquares of sides g_{1}, …,g_{n} 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 |

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### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics

### Cite this

*Australasian Journal of Combinatorics*,

*71*(3), 501-536.

**On a latin square problem of fuchs.** / Colbourn, Charles.

Research output: Contribution to journal › Article

*Australasian Journal of Combinatorics*, vol. 71, no. 3, pp. 501-536.

}

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 -