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Minimum degree thresholds for bipartite graph tiling

For any bipartite graph $H$, we determine a minimum degree threshold for a balanced bipartite graph $G$ to contain a perfect $H$-tiling. We show that this threshold is best possible up to a constant depending only on $H$. Additionally, we prove a corresponding minimum degree threshold to guarantee that $G$ has an $H$-tiling missing only a constant number of vertices. Our threshold for the perfect tiling 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 $χ_{cr}(H)$. Our results answer two questions of Zhao. They can be viewed as bipartite analogs to the results of Kuhn and Osthus and of Shokoufandeh and Zhao.