Vertex degree sums for rainbow matchings in 3-uniform hypergraphs

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 $\hat{I}´>0$ 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 $de{g}_{F_i}(u)+de{g}_{F_i}(v)>\left(\frac{2}{3}+\hat{I}´\right)n^2$ 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.

page 11 1 figure. arXiv admin note: text overlap with arXiv:2004.12558 by other authors