Representing Graphs by Polygons with Edge Contacts in 3D∗

1 A graph has an edge-contact representation with polygons if its vertices can be represented 2 by interior-disjoint polygons such that two polygons share a common edge if and only if the 3 corresponding vertices are adjacent. In this work we study representations in 3D. 4 We show that every graph has an edge-contact representation with polygons in 3D, while this 5 is not the case if we additionally require that the polygons are convex. In particular, we show that 6 every graph containing K5 as a subgraph does not admit a representation with convex polygons. 7 On the other hand, K4,4 admits such a representation, and so does every Kn where every edge 8 is subdivided by a vertex. We also construct an infinite family (Gn) of graphs where Gn has n 9 vertices, 6n−o(n) edges, and admits an edge-contact representation with convex polygons in 3D. 10