Codegree Conditions for Tiling Complete k-Partite k-Graphs and Loose Cycles

摘要

Given two $k$-graphs ($k$-uniform hypergraphs) $F$ and $H$, a perfect$F$-tiling (or an $F$-factor) in $H$ is a set of vertex disjoint copies of $F$that together cover the vertex set of $H$. For all complete $k$-partite$k$-graphs $K$, Mycroft proved a minimum codegree condition that guarantees a$K$-factor in an $n$-vertex $k$-graph, which is tight up to an error term$o(n)$. In this paper we improve the error term in Mycroft's result to asub-linear term that relates to the Tur'an number of $K$ when the differencesof the sizes of the vertex classes of $K$ are co-prime. Furthermore, we find aconstruction which shows that our improved codegree condition is asymptoticallytight in infinitely many cases thus disproving a conjecture of Mycroft. Atlast, we determine exact minimum codegree conditions for tiling $K^{(k)}(1,dots, 1, 2)$ and tiling loose cycles thus generalizing results of Czygrinow,DeBiasio, and Nagle, and of Czygrinow, respectively.