  We characterize Borel line graphs in terms of 10 forbidden induced subgraphs,
namely the 9 finite graphs from the classical result of Beineke together with a
10th infinite graph associated to the equivalence relation $\mathbb{E}_0$ on
the Cantor space. As a corollary, we prove a partial converse to the
Feldman--Moore theorem, which allows us to characterize all locally countable
Borel line graphs in terms of their Borel chromatic numbers.
