Rainbow perfect matchings in 3-partite 3-uniform hypergraphs
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910788845830144 |
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| author | Lu, Hongliang Wang, Yan |
| author_facet | Lu, Hongliang Wang, Yan |
| contents | Let $m,n,r,s$ be nonnegative integers such that $n\ge m=3r+s$ and $1\leq s\leq 3$. 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 $F_1,\ldots, F_n$ 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 $δ(n,r,s)+1$ for $i \in [n]$ then $\{F_1,\ldots,F_n\}$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_08523 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Rainbow perfect matchings in 3-partite 3-uniform hypergraphs Lu, Hongliang Wang, Yan Combinatorics Let $m,n,r,s$ be nonnegative integers such that $n\ge m=3r+s$ and $1\leq s\leq 3$. 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 $F_1,\ldots, F_n$ 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 $δ(n,r,s)+1$ for $i \in [n]$ then $\{F_1,\ldots,F_n\}$ 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. |
| title | Rainbow perfect matchings in 3-partite 3-uniform hypergraphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2408.08523 |