Dot product representations of planar graphs

A graph G is a k -dot product graph if there exists a vector labelling u:V(G)→R k such that u(i) T u(j)≥1 if and only if ij∈E(G) . Fiduccia, Scheinerman, Trenk and Zito [Discrete Math., 1998] asked whether every planar graph is a 3 -dot product graph. We show that the answer is "no". On the other hand, every planar graph is a 4 -dot product graph. We also answer the corresponding questions for planar graphs of prescribed girth and for outerplanar graphs.