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

Abstract

AbstractGiven twok-graphs (k-uniform hypergraphs)FandH, a perfectF-tiling (orF-factor) inHis a set of vertex-disjoint copies ofFthat together cover the vertex set ofH. For all completek-partitek-graphsK, Mycroft proved a minimum codegree condition that guarantees aK-factor in ann-vertexk-graph, which is tight up to an error termo(n). In this paper we improve the error term in Mycroft’s result to a sublinear term that relates to the Turán number ofKwhen the differences of the sizes of the vertex classes ofKare 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. Finally, we determine exact minimum codegree conditions for tilingK(k)(1, … , 1, 2) and tiling loose cycles, thus generalizing the results of Czygrinow, DeBiasio and Nagle, and of Czygrinow, respectively.