Vertex degree sums for rainbow matchings in 3-uniform hypergraphs
Haorui Liu, Mei Lu, Yan Wang, Yi Zhang
TL;DR
We prove an asymptotic rainbow analogue of a Dirac-type degree-sum condition for perfect matchings in $3$-uniform hypergraphs. By translating a rainbow matching problem for the family $\mathscr{F}=\{F_1,\dots, F_{n/3}\}$ into a balanced $(1,3)$-partite $4$-graph $H_{1,3}(\mathscr{F})$, we combine absorbing methods, fractional perfect matching results, and almost-perfect matchings to conclude a rainbow perfect matching under the condition $\sigma_2(F_i) > \left(\frac{2}{3}+\delta\right)n^2$ for all $i$ and large $n$ divisible by $3$. The construction yields an asymptotically tight bound and extends previous Dirac/Ore-type results to the rainbow setting. The techniques—absorbing lemmas, stability-based fractional arguments, and probabilistic coverings—advance rainbow matching theory for $3$-uniform hypergraphs and related $(1,3)$-partite $4$-graphs.
Abstract
Let $n \in 3\mathbb{Z}$ be sufficiently large. Zhang, Zhao and Lu proved that if $H$ is a 3-uniform hypergraph with $n$ vertices and no isolated vertices, and if $deg(u)+deg(v) > \frac{2}{3}n^2 - \frac{8}{3}n + 2$ for any two vertices $u$ and $v$ that are contained in some edge of $H$, then $ H $ admits a perfect matching. In this paper, we prove that the rainbow version of Zhang, Zhao and Lu's result is asymptotically true. More specifically, let $δ> 0$ and $ F_1, F_2, \dots, F_{n/3} $ be 3-uniform hypergraphs on a common set of $n$ vertices. For each $ i \in [n/3] $, suppose that $F_i$ has no isolated vertices and $deg_{F_i}(u)+deg_{F_i}(v) > \left( \frac{2}{3} + δ\right)n^2$ holds for any two vertices $u$ and $v$ that are contained in some edge of $F_i$. Then $ \{ F_1, F_2, \dots, F_{n/3} \} $ admits a rainbow matching. Note that this result is asymptotically tight.