Hamiltonian Zipper Paths Overview

• Hamiltonian zipper paths are combinatorial-geometric structures that integrate polyhedral unfolding, alternating color cycles, and recursive tiling into a unified framework.
• They support efficient algorithms for edge-unfolding, perimeter-halving zipping, and self-avoiding recursive constructions in varied geometric settings.
• Open challenges include fully characterizing zipper pairs and deriving closed-form enumerations, which drive ongoing research in discrete and computational geometry.

A Hamiltonian zipper path is a combinatorial and geometric structure exhibiting deep interconnections between path enumeration, polyhedral unfolding, recursive symmetry, and discrete geometry. The notion encompasses multiple related frameworks: edge-unfoldings of convex polyhedra that "zipper-refold" to compact nets, alternating color Hamiltonian paths with crossing minimization in planar graphs, and recursive self-avoiding curves in regular tiling lattices, notably in the context of the generalized Sierpiński Arrowhead curve. The following sections present a rigorous review of Hamiltonian zipper paths from the perspectives developed in fundamental references (O'Rourke, 2010, Claverol et al., 2016), and (Kaszanyitzky, 2017).

1. Formal Definitions and Core Models

Let $G = (V,E)$ denote the 1-skeleton of either a convex polyhedron $Q \subset \mathbb{R}^3$ or a planar geometric graph. A Hamiltonian zipper path $P = (v_1, \dots, v_n)$ is a simple path visiting all vertices exactly once, where each consecutive pair $(v_i, v_{i+1})$ is either an edge in $E$ or forms a valid step in the combinatorial configuration of interest. For polyhedral unfoldings, $P$ is realized along edges of $Q$, producing a simply-connected polygonal net $N \subset \mathbb{R}^2$ upon cutting, with a boundary $\partial N$ parametrized by arclength.

In recursive tiling (as for the Sierpiński Arrowhead), the context is the inscribed centroid graph of up-facing dark tiles in a partitioned triangular grid, with $T_n = n(n+1)/2$ centroids for order $n$. Here, a path $h_n$ encodes its stepwise direction via an absolute direction code $d \in \{0,1,2,3,4,5\}$ corresponding to $d\pi/3$ radians, and is self-avoiding under "no back" and forbidden turn constraints (Kaszanyitzky, 2017). In alternating coloring setups, the paths are defined on bicolored point sets $S = R \cup B$ and must alternate between red and blue vertices, forming either a cycle or a path (Claverol et al., 2016).

2. Polyhedral Zipper-Unfoldings and Flat Zipper Pairs

For convex polyhedra, a Hamiltonian zipper path determines an edge-cutting sequence producing a planar net $N$ with boundary of length $L$. A zipper-folding is a perimeter-halving gluing of boundary points: after normalizing $L=1$, select an offset $x \in [0, 1/2]$, parametrize $\partial N$ as $p(s)$ with $s \in [0,1)$, and identify $p(s) \leftrightarrow p(s+1/2)$ for $s \in [x, x+1/2)$, modulo cyclic shifts. Alexandrov's gluing theorem ensures existence and uniqueness of a convex polyhedral surface under these constraints.

Flat zipper-pairs are those for which the glued surface is a doubly-covered parallelogram: opposite intervals on $\partial N$ have equal length, and all interior angles pair to sum $\pi$. Algorithmically, one need only check $O(n)$ candidates for $x$, specifically those aligning reflex and convex net vertices (O'Rourke, 2010).

A detailed inventory for the Platonic solids shows that the tetrahedron, cube, octahedron, and icosahedron each admit Hamiltonian zipper-unfoldings to flat parallelogram nets—with explicit examples and net parameters—while the dodecahedron is strictly zip-rigid, admitting only trivial perimeter-matching and no flat zipper-mate due to angle constraints at unfolding endpoints (three $108^\circ$ dihedral angles, yielding an interior $324^\circ$) (O'Rourke, 2010).

3. Hamiltonian Alternating Zipper Paths in Planar and Convex Sets

In bicolored planar point sets, a Hamiltonian alternating zipper path (abbreviated as 1-PHAP or 1-PHAC for paths and cycles, respectively) is a path or cycle that alternates color at each step and visits every point exactly once. These are realized in geometric graphs where each edge is straight and, for 1-plane property, each edge may be crossed at most once.

Key existence results are:

• For $|R| = |B| = n$, any pair of boundary points $(r, b)$ of opposite color not in "special configuration" defines a 1-plane alternating Hamiltonian path with at most $n - r(S)$ crossings, where $r(S)$ is the number of red (resp. blue) runs along the convex hull.
• For convex position, any minimum-crossing alternating path or cycle is necessarily 1-plane, and a linear-time algorithm ("$J$-PAIRS") matches each red point to its balancing blue and constructs the path by alternating these matches, with minimum crossings $n - r(S)$ (Claverol et al., 2016).

Dynamic programming yields $O(n^2)$ algorithms for arbitrary endpoints on the convex boundary, leveraging the forced structure of optimum paths via breakpoints determined by balancing partners.

4. Recursive Hamiltonian Zipper Paths and Sierpiński Arrowhead Generalization

In the recursive tiling framework, Hamiltonian zipper paths manifest as generator curves for self-avoiding recursive paths over the tiled grid. There are two fundamental recursive constructions:

• Node-rewriting (NR): A generator string $w_n^1$ defines well-formed paths on the centroid grid, avoiding "wrong" $120^\circ$ turns (modulo $6$) after even and odd steps.
• Edge-rewriting (ER): $s_n^1$ encodes a tiling-path on the full grid, and each even/odd code is locally rewritten by explicit modular addition rules emulating, within each triangle, the $T_n$-edge path structure.

A canonical bijection (via explicit code conversion tables) maps NR to ER paths, preserving self-avoiding and simplicity properties through recursive steps (Kaszanyitzky, 2017). Both path types can be rendered as Lindenmayer-system strings by further code-table translations.

A table of path counts for small $n$ demonstrates rapid growth:

$n$	$|H_n|$ (All Hamiltonian Paths)	$|W_n|=|S_n|$ (Well-formed/ER-paths)
2	1	1
3	2	2
4	10	4
5	92	16
6	1852	68
7	78,032	464
8	6,846,876	3,828
9	1,255,156,712	44,488

No closed-form is known; well-formed path counts correspond to the enumeration of zig-zag-free Hamiltonian tours on the graph of centroids for order $n$ (Kaszanyitzky, 2017).

5. Algorithms and Transformations

Efficient explicit procedures characterize the construction and verification of Hamiltonian zipper paths in each domain:

• For polyhedra, the process involves edge-unfolding, boundary parametrization, perimeter-halving zipping, and Alexandrov validation of the refolding (O'Rourke, 2010).
• For bicolored convex positions, linear algorithms build optimum 1-PHAC and 1-PHAP structures, with $J$-PAIRS computably matching partners using dual stack traversals of the boundary sequence (Claverol et al., 2016). Dynamic programming extends these methods to arbitrary endpoint pairs.
• Triangular grid recursions are implemented by expanding each generator string digit as per modular rules (either NR or ER), with forbidden local turns ensuring self-avoidance. Explicit transformation tables allow conversion between generator types and L-system format (Kaszanyitzky, 2017).

The inductive structure of these constructions ensures that recursive self-avoiding properties are preserved and all transformations are invertible and unambiguous.

6. Limitations, Rigidity, and Open Problems

For convex polyhedra, not all solids admit nontrivial flat zipper pairs. The dodecahedron is a canonical example of zip-rigidity due to the angular deficit at endpoints after unfolding; the only available zipper simply recovers the original solid (O'Rourke, 2010). Characterizing which pairs of convex polyhedra form zipper pairs remains open, as does extending this analysis to Archimedean, Catalan, and higher-genus solids.

In alternating color settings, configurations called "special" can prevent the existence of 1-PHAPs under the given constraints; the full characterization of such obstructive cases is unresolved (Claverol et al., 2016).

Enumeration of well-formed Hamiltonian zipper paths in recursive tilings lacks a closed formula, and precise asymptotics remain open (Kaszanyitzky, 2017).

7. Connections and Broader Context

Hamiltonian zipper paths synthesize concepts from discrete geometry, combinatorial topology, computational geometry, algebraic recursive systems, and tiling theory. They provide compact, highly symmetric representations of structures such as Platonic solids and recursive tilings like the Sierpiński Arrowhead. The linkage among seemingly disparate topics—unfolding of 3D polyhedra, minimum crossing alternator cycles, and generalized fractal curves—underscores the unifying potential of the Hamiltonian zipper paradigm for both theoretical advances and algorithmic developments.

References:

• (O'Rourke, 2010) Flat Zipper-Unfolding Pairs for Platonic Solids.
• (Claverol et al., 2016) On Hamiltonian alternating cycles and paths.
• (Kaszanyitzky, 2017) The generalized Sierpiński Arrowhead Curve.