Local rainbow colorings of hypergraphs

Abstract

In this paper, we generalize the concepts related to rainbow coloring to hypergraphs. Specifically, an $(n,r,H)$-local coloring is defined as a collection of $n$ edge-colorings, $f_v: E(K^{(r)}_n) \rightarrow [k]$ for each vertex $v$ in the complete $r$-uniform hypergraph $K^{(r)}_n$, with the property that for any copy $T$ of $H$ in $K^{(r)}_n$, there exists at least one vertex $u$ in $T$ such that $f_u$ provides a rainbow edge-coloring of $T$ (i.e., no two edges in $T$ share the same color under $f_u$). The minimum number of colors required for this coloring is denoted as the local rainbow coloring number $C_r(n, H)$. We first establish an upper bound of the local rainbow coloring number for $r$-uniform hypergraphs $H$ consisting of $h$ vertices, that is, $C_r(n, H)= O\left( n^{\frac{h-r}{h}} \cdot h^{2r + \frac{r}{h}} \right)$. Furthermore, we identify a set of $r$-uniform hypergraphs whose local rainbow coloring numbers are bounded by a constant. A notable special case indicates that $C_3(n,H) \leq C(H)$ for some constant $C(H)$ depending only on $H$ if and only if $H$ contains at most 3 edges and does not belong to a specific set of three well-structured hypergraphs, possibly augmented with isolated vertices. We further establish two 3-uniform hypergraphs $H$ of particular interest for which $C_3(n,H) = n^{o(1)}$. Regarding lower bounds, we demonstrate that for every $r$-uniform hypergraph $H$ with sufficiently many edges, there exists a constant $b = b(H) > 0$ such that $C_r(n,H) = Ω(n^b)$. Additionally, we obtain lower bounds for several hypergraphs of specific interest.