Side-Contact Representations with Convex Polygons in 3D: New Results for Complete Bipartite Graphs
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A polyhedral surface 𝒞 in ℝ^3 with convex polygons as faces is a side-contact representation of a graph G if there is a bijection between the vertices of G and the faces of 𝒞 such that the polygons of adjacent vertices are exactly the polygons sharing an entire common side in 𝒞. We show that K_3,8 has a side-contact representation but K_3,250 has not. The latter result implies that the number of edges of a graph with side-contact representation and n vertices is bounded by O(n^5/3).
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