Rainbow Matchings in Properly-Colored Hypergraphs

Hao Huang; Tong Li; Guanghui Wang

doi:10.37236/8125

Rainbow Matchings in Properly-Colored Hypergraphs

The Electronic Journal of Combinatorics ◽

10.37236/8125 ◽

2019 ◽

Vol 26 (1) ◽

Author(s):

Hao Huang ◽

Tong Li ◽

Guanghui Wang

Keyword(s):

Rainbow Matching ◽

Uniform Hypergraphs

A hypergraph $H$ is properly colored if for every vertex $v\in V(H)$, all the edges incident to $v$ have distinct colors. In this paper, we show that if $H_{1}, \ldots, H_{s}$ are properly-colored $k$-uniform hypergraphs on $n$ vertices, where $n\geq3k^{2}s$, and $e(H_{i})>{{n}\choose {k}}-{{n-s+1}\choose {k}}$, then there exists a rainbow matching of size $s$, containing one edge from each $H_i$. This generalizes some previous results on the Erdős Matching Conjecture.

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The Maximum Number of Cliques in Hypergraphs without Large Matchings

The Electronic Journal of Combinatorics ◽

10.37236/9604 ◽

2020 ◽

Vol 27 (4) ◽

Author(s):

Erica L.L. Liu ◽

Jian Wang

Keyword(s):

Sufficient Condition ◽

Matching Number ◽

Uniform Hypergraph ◽

Rainbow Matching ◽

Uniform Hypergraphs ◽

Large N ◽

Vertex Set

Let $[n]$ denote the set $\{1, 2, \ldots, n\}$ and $\mathcal{F}^{(r)}_{n,k,a}$ be an $r$-uniform hypergraph on the vertex set $[n]$ with edge set consisting of all the $r$-element subsets of $[n]$ that contains at least $a$ vertices in $[ak+a-1]$. For $n\geq 2rk$, Frankl proved that $\mathcal{F}^{(r)}_{n,k,1}$ maximizes the number of edges in $r$-uniform hypergraphs on $n$ vertices with the matching number at most $k$. Huang, Loh and Sudakov considered a multicolored version of the Erd\H{o}s matching conjecture, and provided a sufficient condition on the number of edges for a multicolored hypergraph to contain a rainbow matching of size $k$. In this paper, we show that $\mathcal{F}^{(r)}_{n,k,a}$ maximizes the number of $s$-cliques in $r$-uniform hypergraphs on $n$ vertices with the matching number at most $k$ for sufficiently large $n$, where $a=\lfloor \frac{s-r}{k} \rfloor+1$. We also obtain a condition on the number of $s$-clques for a multicolored $r$-uniform hypergraph to contain a rainbow matching of size $k$, which reduces to the condition of Huang, Loh and Sudakov when $s=r$.

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Rainbow matchings for 3-uniform hypergraphs

Journal of Combinatorial Theory Series A ◽

10.1016/j.jcta.2021.105489 ◽

2021 ◽

Vol 183 ◽

pp. 105489

Author(s):

Hongliang Lu ◽

Xingxing Yu ◽

Xiaofan Yuan

Keyword(s):

Uniform Hypergraphs

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Hypercycle Systems of 5-Cycles in Complete 3-Uniform Hypergraphs

Mathematics ◽

10.3390/math9050484 ◽

2021 ◽

Vol 9 (5) ◽

pp. 484

Author(s):

Anita Keszler ◽

Zsolt Tuza

Keyword(s):

Element Subset ◽

Uniform Hypergraphs ◽

Vertex Set ◽

Recursive Constructions

In this paper, we consider the problem of constructing hypercycle systems of 5-cycles in complete 3-uniform hypergraphs. A hypercycle system C(r,k,v) of order v is a collection of r-uniform k-cycles on a v-element vertex set, such that each r-element subset is an edge in precisely one of those k-cycles. We present cyclic hypercycle systems C(3,5,v) of orders v=25,26,31,35,37,41,46,47,55,56, a highly symmetric construction for v=40, and cyclic 2-split constructions of orders 32,40,50,52. As a consequence, all orders v≤60 permitted by the divisibility conditions admit a C(3,5,v) system. New recursive constructions are also introduced.

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