Hamiltonian Sets of Polygonal Paths in Assembly 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.

Figures (9)

• Figure 1: Some diagrams for a rigid vertex $v$ of degree 4.
• Figure 2: Consecutevily drawing transverse path $v_0\,e_0\,v_1\,e_1\,v_1\,e_2\,v_2\,e_3\,v_3\,e_4\,v_3\,e_5\,v_2\,e_6\,v_4$.
• Figure 3: An example of Hamiltonian set of polygonal paths with three paths: $\{v_1 e_1 v_2 e_{15} v_7, \, v_5 e_4 v_4 e_{13} v_8 e_{10} v_3, \,v_6\}$. Note that $v_6$ is a singleton.
• Figure 4: How the tangled cord $TC_{n}$ is inductively constructed from $TC_{n-1}$. The visualization provides the 'correct' cyclic order of half-edges in each rigid vertex.
• Figure 5: Enumerated edges in a graph with the assembly word $112323$

Theorems & Definitions (25)

• Conjecture 1.1
• Definition 2.1
• Definition 2.2
• Example 2.3
• Definition 2.4
• Remark 2.5
• Theorem 3.1
• Definition 3.2
• Example 3.3
• Definition 3.4