Hamiltonian Sets of Polygonal Paths in Assembly Graphs

TL;DR

This paper characterizes assembly graphs with the maximum number of Hamiltonian polygonal path sets, proving that only special graphs called tangled cords achieve this maximum, which relates to Fibonacci numbers.

Contribution

It provides four equivalent combinatorial conditions for identifying assembly graphs with maximum Hamiltonian polygonal path sets, confirming a Fibonacci-based conjecture.

Findings

Maximum number of Hamiltonian sets equals F_{2n+1}-1

Maximum is achieved only by tangled cords

Four equivalent conditions characterize these graphs

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 its size. These conditions serve to prove the conjecture that such maximum, which is equal to $F_{2n+1}-1$, where $F_k$ denotes the $k$th Fibonacci number, is achieved only for special assembly graphs, called tangled cords.