Minimum vertex degree thresholds for tiling complete 3-partite 3-graphs

Abstract

Given positive integers $a\le b\le c$, let $K_{a,b,c}$ be the complete 3-partite 3-uniform hypergraph with three parts of sizes $a,b,c$. Let $H$ be a 3-uniform hypergraph on $n$ vertices where $n$ is divisible by $a+b+c$. We asymptotically determine the minimum vertex degree of $H$ that guarantees a perfect $K_{a,b,c}$-tiling, that is, a spanning subgraph of $H$ consisting of vertex-disjoint copies of $K_{a,b,c}$. This partially answers a question of Mycroft, who proved an analogous result with respect to codegree for $r$-uniform hypergraphs for all $r\ge 3$. Our proof uses a lattice-based absorbing method, the concept of fractional tiling, and a recent result on shadows for 3-graphs.

21 pages, 1 figure

References in corpus (3)

• Further applications of the Container Method
• Minimum vertex degree conditions for loose Hamilton cycles in $3$-uniform hypergraphs
• Recent advances on Dirac-type problems for hypergraphs

Cited by in corpus (11)

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• Minimum codegree threshold for $C_6^3$-factors in $3$-uniform Hypergraphs
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• Codegree conditions for tilling balanced complete $3$-partite $3$-graphs and generalized 4-cycles
• Triangle-degrees in graphs and tetrahedron coverings in 3-graphs
• On Perfect Matchings and tilings in uniform Hypergraphs
• Tiling multipartite hypergraphs in Quasi-random Hypergraphs
• Codegree thresholds for covering 3-uniform hypergraphs
• Embedding clique-factors in graphs with low $\ell$-independence number
• Exact minimum codegree thresholds for $K_4^{-}$-covering and $K_5^{-}$-covering