Rainbow perfect matchings in 3-partite 3-uniform hypergraphs

Abstract

Let $m,n,r,s$ be nonnegative integers such that $n\ge m=3r+s$ and $1\le s\le 3$. Let \[δ(n,r,s)=\left\{\begin{array}{ll} n^2-(n-r)^2 &\text{if}\ s=1 , \\[5pt] n^2-(n-r+1)(n-r-1) &\text{if}\ s=2,\\[5pt] n^2 - (n-r)(n-r-1) &\text{if}\ s=3. \end{array}\right.\] We show that there exists a constant $n_0>0$ such that if $F_1,\dots ,F_n$ are 3-partite 3-graphs with $n\ge n_0$ vertices in each partition class and minimum vertex degree of $F_i$ is at least $\hat{I}´(n,r,s)+1$ for $i\in [n]$ then $\{F_1,\dots ,F_n\}$ admits a rainbow perfect matching. This generalizes a result of Lo and Markström on the vertex degree threshold for the existence of perfect matchings in 3-partite 3-graphs. In this proof, we use a fractional rainbow matching theory obtained by Aharoni et al. to find edge-disjoint fractional perfect matching.