Label vertices by triples of bits and join pairs differing in exactly one bit. Draw the four vertices with first bit zero as an outer square, and the four with first bit one as an inner square, in the same cyclic order for the last two bits. Connect corresponding corners by four nonintersecting spokes. Every cube edge is present and there are no crossings, giving the required planar embedding.
Figure 1.
Planar embedding of the cube graph with binary vertex labels
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