Year: 2024
Type: preprint
Abstract: 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
Source: arXiv (Cornell University)
Authors: Liu, Xizhi, Ma, Jie, Wang, Tianhen, Zhu, Tianming
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: G1330075977, G3857364911
OpenAlex ID: W4402955066