A digraph admits a completely reachable road coloring iff it is strongly connected, aperiodic, and every vertex subset has at least as many in-neighbors as vertices; the fixed-alphabet version is claimed NP-complete, but the proof has a flaw.
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Completely Reachable Road Coloring
A digraph admits a completely reachable road coloring iff it is strongly connected, aperiodic, and every vertex subset has at least as many in-neighbors as vertices; the fixed-alphabet version is claimed NP-complete, but the proof has a flaw.