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 c n 2 − 2 β 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.