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, K_{s, t}) exactly for all s?t and suffi‐ciently large n. In this article 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 χ_cr(H), while the threshold for the almost perfect tiling only depends on χ_cr(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]. © 2011 Wiley Periodicals, Inc. J Graph Theory