On the existence of a rainbow 1-factor in proper coloring of K_{rn}^{(r)}

TL;DR

This paper extends previous results by proving that any proper coloring of a complete uniform hypergraph $K_{rn}^{(r)}$ with $r extgreater=2$ and $n extgreater=3$ guarantees the existence of a rainbow 1-factor, broadening the understanding of rainbow structures in hypergraphs.

Contribution

The authors generalize prior work by showing that proper colorings, not just 1-factorizations, ensure rainbow 1-factors in complete uniform hypergraphs.

Findings

Proper colorings guarantee rainbow 1-factors in $K_{rn}^{(r)}$.

The result applies for all $r extgreater= 2$ and $n extgreater= 3$.

Extends previous theorems from 1-factorizations to proper colorings.

Abstract

El-Zanati et al proved that for any 1-factorization $\mathcal{F}$ of the complete uniform hypergraph $\mathcal {G}=K_{rn}^{(r)}$ with $r\geq 2$ and $n\geq 3$, there is a rainbow 1-factor. We generalize their result and show that in any proper coloring of the complete uniform hypergraph $\mathcal {G}=K_{rn}^{(r)}$ with $r\geq 2$ and $n\geq 3$, there is a rainbow 1-factor.