Constructing an icosahedron from cardboard and string (Q2772827)

The authors discuss a relation between an Eulerian circuit of a graph and cardboard and string models of polyhedra. They prove that there exists a twist-free Eulerian circuit of a connected graph \(G\) with partition function \(F_{v}\) for all \(v\in G\) if and only if \(|F^{-1}_{v}(0)|= |F^{-1}_{v}(1)|\) for all \(v\in G\). Their result shows that twist-free cardboard and string models are possible for many polyhedra, such as the icosahedron, the snub cuboctahedron and the snub icosahedron.