Abstract We provide four equivalent combinatorial conditions for a simple assembly graph (rigid vertex graph where all vertices are of degree 1 or 4) to have the largest number of Hamiltonian sets of polygonal paths relative to its size. These conditions serve to prove the conjecture that such a maximum, which is equal to F₂₍+₁-1, where Fₖ denotes the k th Fibonacci number, is achieved only for special assembly graphs, called tangled cords.
Guterman et al. (2026) studied this question.