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

Abstract

Given positive integers a≤b≤c, let Ka,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 Ka,b,c-tiling, that is, a spanning subgraph of H consisting of vertex-disjoint copies of Ka,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≥3. Our proof uses a lattice-based absorbing method, the concept of fractional tiling, and a recent result on shadows for 3-graphs.