The Turán number of sparse spanning graphs

For a graph H, the extremal number ex(n,H) is the maximum number of edges in a graph of order n not containing a subgraph isomorphic to H. Let δ(H) > 0 and ∆(H) denote the minimum degree and maximum degree of H, respectively. We prove that for all n sufficiently large, if H is any graph of order n with ∆(H) ≤ √ n/200, then ex(n,H) = ( n−1 2 ) +δ(H)−1. The condition on the maximum degree is tight up to a constant factor. This generalizes a classical result of Ore for the case H = Cn, and resolves, in a strong form, a conjecture of Glebov, Person, and Weps for the case of graphs. A counter-example to their more general conjecture concerning the extremal number of bounded degree spanning hypergraphs is also given.