Codegree conditions for tiling complete \(k\)-partite \(k\)-graphs and loose cycles

Abstract

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 a sub-linear term that relates to the Tur\'an number of \(K\) when the differences of the sizes of the vertex classes of \(K\) are co-prime. Furthermore, we find a construction which shows that our improved codegree condition is asymptotically tight in infinitely many cases thus disproving a conjecture of Mycroft. At last, 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.