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We show that every graph of maximum degree 3 can be represented as theintersection graph of axis parallel boxes in three dimensions, that is, everyvertex can be mapped to an axis parallel box such that two boxes intersect ifand only if their corresponding vertices are adjacent. In fact, we construct arepresentation in which any two intersecting boxes just touch at theirboundaries. Further, this construction can be realized in linear time.

Representing a cubic graph as the intersection graph of axis-parallel boxes in three dimensions