Graphs with large maximum degree containing no odd cycles of a given length

Let us write f(n,Δ;C2k+1) for the maximal number of edges in a graph of order n and maximum degree Δ that contains no cycles of length 2k+1. For n2⩽Δ⩽n−k−1 and n sufficiently large we show that f(n,Δ;C2k+1)=Δ(n−Δ), with the unique extremal graph a complete bipartite graph.