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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2605.28027 |
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| _version_ | 1866910264733990912 |
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| author | Bean, Richard Cavenagh, Nicholas |
| author_facet | Bean, Richard Cavenagh, Nicholas |
| contents | A defining set of a Latin square is a partially filled-in Latin square which completes to no other Latin square of the same order. We introduce the concept of a $k$-strong defining set, in which if less than $k$ entries are deleted, the property of being a defining set is retained. Equivalently, a $k$-strong defining set intersects every Latin trade in the Latin square at least $k$ times. In the addition table for integers modulo $n$, when $n$ is even we determine the minimum size of a $k$-strong defining set for any $k$. For odd $n$ we give a construction for a minimally $2$-strong defining set. We furthermore give computational results for Latin squares of small orders. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_28027 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Defining sets which intersect each Latin trade at least twice Bean, Richard Cavenagh, Nicholas Combinatorics 05B15 A defining set of a Latin square is a partially filled-in Latin square which completes to no other Latin square of the same order. We introduce the concept of a $k$-strong defining set, in which if less than $k$ entries are deleted, the property of being a defining set is retained. Equivalently, a $k$-strong defining set intersects every Latin trade in the Latin square at least $k$ times. In the addition table for integers modulo $n$, when $n$ is even we determine the minimum size of a $k$-strong defining set for any $k$. For odd $n$ we give a construction for a minimally $2$-strong defining set. We furthermore give computational results for Latin squares of small orders. |
| title | Defining sets which intersect each Latin trade at least twice |
| topic | Combinatorics 05B15 |
| url | https://arxiv.org/abs/2605.28027 |