Uniquely colorable hypergraphs
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929484009046016 |
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| author | Liu, Xizhi Ma, Jie Wang, Tianhen Zhu, Tianming |
| author_facet | Liu, Xizhi Ma, Jie Wang, Tianhen Zhu, Tianming |
| contents | An $r$-uniform hypergraph is uniquely $k$-colorable if there exists exactly one partition of its vertex set into $k$ parts such that every edge contains at most one vertex from each part. For integers $k \ge r \ge 2$, let $Φ_{k,r}$ denote the minimum real number such that every $n$-vertex $k$-partite $r$-uniform hypergraph with positive codegree greater than $Φ_{k,r} \cdot n$ and no isolated vertices is uniquely $k$-colorable. A classic result by of Bollobás\cite{Bol78} established that $Φ_{k,2} = \frac{3k-5}{3k-2}$ for every $k \ge 2$.
We consider the uniquely colorable problem for hypergraphs. Our main result determines the precise value of $Φ_{k,r}$ for all $k \ge r \ge 3$. In particular, we show that $Φ_{k,r}$ exhibits a phase transition at approximately $k = \frac{4r-2}{3}$, a phenomenon not seen in the graph case. As an application of the main result, combined with a classic theorem by Frankl--Füredi--Kalai, we derive general bounds for the analogous problem on minimum positive $i$-degrees for all $1\leq i<r$, which are tight for infinitely many cases. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_01654 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Uniquely colorable hypergraphs Liu, Xizhi Ma, Jie Wang, Tianhen Zhu, Tianming Combinatorics An $r$-uniform hypergraph is uniquely $k$-colorable if there exists exactly one partition of its vertex set into $k$ parts such that every edge contains at most one vertex from each part. For integers $k \ge r \ge 2$, let $Φ_{k,r}$ denote the minimum real number such that every $n$-vertex $k$-partite $r$-uniform hypergraph with positive codegree greater than $Φ_{k,r} \cdot n$ and no isolated vertices is uniquely $k$-colorable. A classic result by of Bollobás\cite{Bol78} established that $Φ_{k,2} = \frac{3k-5}{3k-2}$ for every $k \ge 2$. We consider the uniquely colorable problem for hypergraphs. Our main result determines the precise value of $Φ_{k,r}$ for all $k \ge r \ge 3$. In particular, we show that $Φ_{k,r}$ exhibits a phase transition at approximately $k = \frac{4r-2}{3}$, a phenomenon not seen in the graph case. As an application of the main result, combined with a classic theorem by Frankl--Füredi--Kalai, we derive general bounds for the analogous problem on minimum positive $i$-degrees for all $1\leq i<r$, which are tight for infinitely many cases. |
| title | Uniquely colorable hypergraphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2409.01654 |