Local rainbow colorings of hypergraphs

TL;DR

The paper generalizes rainbow coloring to $r$-uniform hypergraphs by introducing $(n,r,H)$-local colorings and the local rainbow coloring number $C_r(n,H)$. It develops upper bounds via probabilistic methods and the Lovász Local Lemma, establishes a constant-bound dichotomy determined by the 2-locally large property, and proves subpolynomial upper bounds for specific hypergraphs using $(p,q)$-colorings. On the lower-bound side, it shows polynomial growth for cliques and sunflowers and provides detailed bounds for several 3-graphs, including special paths, highlighting the spectrum between polynomial and subpolynomial behavior. Overall, the work reveals rich interactions between hypergraph structure and local rainbow colorings, with connections to Ramsey-type phenomena and subgraph containment, and suggests directions for sharpening bounds and widening the class of hypergraphs with known growth rates.

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.

Figures (5)

• Figure 1: A tight $P_3$, abbreviated as ${\rm TP}_3$.
• Figure 2: A special $P_3$, abbreviated as ${\rm SP}_3$.
• Figure 3: A loose $C_3$, abbreviated as ${\rm LC}_3$.
• Figure 4: A tight $C_3$ with a pendant edge, abbreviated as ${\rm TC}_e$.
• Figure 5: A loose cycle with an added edge $adf$, abbreviated as ${\rm LC_e}$

Theorems & Definitions (38)

• Definition 1.1: zbMATH06056109
• Definition 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Corollary 1.7
• Theorem 1.8
• Theorem 1.9
• Lemma 2.1: alon2016probabilistic Lovász local lemma