Codegree threshold for tiling balanced complete 3-partite 3-graphs and generalized 4-cycles

Given two k-graphs F and H, a perfect F-tiling (also called an F-factor) in H is a set of vertex-disjoint copies of F that together cover the vertex set of H. Let t(k-1)(n, F) be the smallest integer t such that every k-graph H on n vertices with minimum codegree at least t contains a perfect F-tiling. Mycroft (JCTA, 2016) determined the asymptotic values of t(k-1)(n, F) for k-partite k-graphs F and conjectured that the error terms o(n) in t(k-1)(n,F) can be replaced by a constant that depends only on F. In this paper, we determine the exact value of t(2)(n,K-m,m(3)), where K-m(,m)3 (defined by Mubayi and Verstraete, JCTA, 2004) is the 3-graph obtained from the complete bipartite graph K-m,K-m by replacing each vertex in one part by a 2-elements set. Note that K-2,(3)(2)( )is the well known generalized 4-cycle C-4(3) (the 3-graph on six vertices and four distinct edges A,B,C,D with A boolean OR B = A boolean OR D and A boolean AND B = C boolean AND D = theta). The result confirms Mycroft's conjecture for K-m,m(3). Moreover, we improve the error term o(n) to a sub-linear term when F = K-3(m) and show that the sub-linear term is tight for K-3(2), where K-3(m) is the complete 3-partite 3-graph with each part of size m.