Minimum degree thresholds for bipartite graph tiling

Given a bipartite graph H and a positive integer n such that v(H) divides 2n, we define the minimum degree threshold for bipartite H-tiling, δ2(n,H), as the smallest integer k such that every bipartite graph G with n vertices in each partition and minimum degree δ(G) ≥ k contains a spanning subgraph consisting of vertex-disjoint copies of H. Zhao, Hladký-Schacht, Czygrinow-DeBiasio determined δ2(n,Ks,t) exactly for all s ≤ t and sufficiently large n. In this paper we determine δ2(n,H), up to an additive constant, for all bipartite H and sufficiently large n. Additionally, we give a corresponding minimum degree threshold to guarantee that G has an H-tiling missing only a constant number of vertices. Our δ2(n,H) depends on either the chromatic number χ(H) or the critical chromatic number χcr(H) while the threshold for the almost perfect tiling only depends on χcr(H). These results can be viewed as bipartite analogs to the results of Kuhn and Osthus [Combinatorica 29 (2009), 65-107] and of Shokoufandeh and Zhao [Rand. Struc. Alg. 23 (2003), 180-205].