 That is, the vertices and edges of the Császár polyhedron form a complete graph.If the boundary of a polyhedron with "v" vertices forms a surface with "h" holes, in such a way that every pair of vertices is connected by an edge, it follows by some manipulation of the Euler characteristic that
This equation is satisfied for the tetrahedron with "h" = 0 and "v" = 4, and for the Császár polyhedron with "h" = 1 and "v" = 7.The next possible solution, "h" = 6 and "v" = 12, would correspond to a polyhedron with 44 faces and 66 edges, but it is not realizable as a polyhedron.