On a problem of Duke–Erdős–Rödl on cycle-connected subgraphs

In this short note, we prove that for β<1/5 every graph G with n vertices and n^2−β edges contains a subgraph G^′ with at least cn^2−2β edges such that every pair of edges in G^′ lie together on a cycle of length at most 8. Moreover edges in G^′ which share a vertex lie together on a cycle of length at most 6. This result is best possible up to the constant factor and settles a conjecture of Duke, Erdős, and Rödl.