With high probability, the cover time of Q_{n,p} is (1+o(1)) (p/(log 2 log(2p/(2p-1)))) n log n for p at least the connectivity threshold.
Bollob´ as, Complete matchings in random subgraphs of the cube
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Cover time of random subgraphs of the hypercube
With high probability, the cover time of Q_{n,p} is (1+o(1)) (p/(log 2 log(2p/(2p-1)))) n log n for p at least the connectivity threshold.