Contact representations of planar graphs with cubes
Stefan Felsner, Mathew C. Francis · 2011
We prove that every planar graph has a representation using axis-parallel cubes in three dimensions in such a way that there is a cube corresponding to each vertex of the planar graph and two cubes have a non-empty intersection if and only if their corresponding vertices are adjacent. Moreover, when two cubes have a non-empty intersection, they just touch each other. This result is a strengthening of a result by Thomassen which states that every planar graph has such a representation using axis-parallel boxes.