Existence of Spanning ℱ-Free Subgraphs with Large Minimum Degree

Let F be a family of graphs and let d be large enough. For every d-regular graph G, we study the existence of a spanning F -free subgraph of G with large minimum degree. This problem is wellunderstood if F does not contain bipartite graphs. Here we provide asymptotically tight results for many families of bipartite graphs such as cycles or complete bipartite graphs. To prove these results, we study a locally injective analogue of the question.