[Paper Review] Combinatorial manifolds are Hamiltonian

Oliver Knill|arXiv (Cornell University)|Jun 17, 2018

Topological and Geometric Data Analysis21 references3 citations

TL;DR

This paper proves that all connected d-graphs—combinatorial manifolds defined inductively by sphere-like unit spheres—are Hamiltonian, extending Whitney's 1931 result. The proof is constructive and shows that finding Hamiltonian cycles in such graphs can be done in polynomial time, offering a significant complexity advantage over general graphs.

ABSTRACT

Extending a theorem of Whitney of 1931 we prove that all connected d-graphs are Hamiltonian for positive d. A d-graph is a type of combinatorial manifold which is inductively defined as a finite simple graph for which every unit sphere is a (d-1)-sphere. A d-sphere is d-graph such that removing one vertex renders the graph contractible. A graph is contractible if there exists a vertex for which the unit sphere and the graph without that vertex are both contractible. These inductive definitions are primed with the assumptions that the empty graph 0 is the (-1)-sphere and that the one-point graph 1 is the smallest contractible graph. The proof is constructive and shows that unlike for general graphs, the complexity of the construction of Hamiltonian cycles in d-graphs is polynomial in the number of vertices of the graph.

Motivation & Objective

• To extend Whitney’s 1931 theorem on Hamiltonian cycles in 2-spheres to higher-dimensional combinatorial manifolds.
• To establish that all connected d-graphs, defined via inductive sphere and contractibility conditions, admit Hamiltonian cycles.
• To demonstrate that the construction of Hamiltonian cycles in d-graphs is polynomial-time, unlike general graphs.
• To provide a constructive proof using inductive definitions of d-spheres and contractible graphs.

Proposed method

• Defining d-graphs inductively: a d-graph is a finite simple graph where every unit sphere is a (d-1)-sphere.
• Defining a d-sphere as a d-graph that becomes contractible upon removal of one vertex.
• Using inductive base cases: the empty graph is the (−1)-sphere, and the one-point graph is the smallest contractible graph.
• Defining contractibility via existence of a vertex whose removal and unit sphere are both contractible.
• Applying induction on dimension d to construct Hamiltonian cycles recursively.
• Ensuring the construction remains polynomial in the number of vertices by maintaining local control over graph structure.

Experimental results

Research questions

• RQ1 Are all connected d-graphs Hamiltonian for d > 0, generalizing Whitney’s result for 2-spheres?
• RQ2 Can a constructive algorithm for Hamiltonian cycles in d-graphs be designed with polynomial time complexity?
• RQ3 How do inductive definitions of spheres and contractibility enable the existence of Hamiltonian cycles?
• RQ4 What is the computational complexity of constructing Hamiltonian cycles in d-graphs compared to general graphs?

Key findings

• All connected d-graphs are Hamiltonian for any positive dimension d.
• The proof is constructive, explicitly building Hamiltonian cycles in the graph.
• The time complexity of constructing a Hamiltonian cycle in a d-graph is polynomial in the number of vertices.
• The construction relies on inductive definitions of d-spheres and contractible graphs using base cases of the empty graph and one-point graph.
• The method avoids exponential blowup by leveraging local topological structure via unit spheres.
• The result generalizes Whitney’s 1931 theorem from 2-spheres to all combinatorial manifolds defined as d-graphs.