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We prove that planar graphs have O(log 4 n) queue number, thus improving upon the previous O( √ n) upper bound. Consequently, planar graphs admit 3D straight-line crossingfree grid drawings in O(n log c n) volume, for some constant c, thus improving upon the previous O(n 3/2 ) upper bound. p,q such that a ≺ b. Then, a is inserted into L ′ p,q before b. Proof: The proof easily descends from the fact that, when a vertex is inserted into a list L ′ p,q , it is appended to such a list. We now studydoi:10.1109/focs.2010.42 dblp:conf/focs/BattistaFP10 fatcat:e2gzytxcj5e65nbdq2vowaqld4