Combinatorial manifolds are Hamiltonian (1806.06436v1)

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.

• The paper establishes a constructive method proving that every connected d-graph admits a Hamiltonian cycle.
• It extends Whitney's theorem using an inductive approach on combinatorial manifolds with a polynomial complexity guarantee.
• Implications include advances in computational topology and computer graphics, enabling efficient mesh traversal on complex surfaces.

Combinatorial Manifolds are Hamiltonian: A Detailed Overview

In a compelling paper, Oliver Knill extends the classic theorem by Whitney, asserting the Hamiltonian nature of a specific class of graphs termed "combinatorial manifolds" or "$d$-graphs" for dimensions $d \geq 1$. This paper establishes a constructive methodology, ensuring that each connected $d$-graph possesses a Hamiltonian cycle and that the construction complexity is polynomial with respect to the number of vertices in the graph.

Key Definitions and Previous Work

The paper builds upon foundational work in the paper of Hamiltonian cycles. Whitney's theorem from 1931 demonstrated that certain planar graphs admit Hamiltonian cycles. The essence of a Hamiltonian cycle is that it visits every vertex exactly once, forming a closed loop. This concept was extensively studied in graphs that can be derived from triangulating spheres, revisited by graph theorists like Bill Tutte, who extended Whitney's work to 4-connected planar graphs.

Knill approaches the problem using $d$-graphs, which are inductively defined combinatorial manifolds where each unit sphere forms a $(d-1)$-sphere. For example, a 2-sphere is a graph where each vertex's neighborhood forms a 1-dimensional cycle, like a triangle. The contractibility condition—an essential concept for defining these graphs—ensures that removing a specific vertex results in a graph that collapses to a single point.

Constructive Proof and Complexity

Central to Knill's results is the constructive nature of the proof, indicating that $d$-graphs inherently admit Hamiltonian cycles. The construction is efficient, with complexity polynomial in the graph's size, contrasting with the general NP-completeness of the Hamiltonian cycle problem in arbitrary graphs. This complexity result highlights the unique properties of $d$-graphs, significantly simplifying the problem from a computational perspective.

The proof strategy involves an inductive approach on dimension $d$, leveraging the graph theoretical notion of generalized $d$-graphs, which include, aside from standard $d$-graphs, cases where all unit spheres are either spheres, balls, or simplices, with certain adjacency conditions near the boundary. This allows for addressing more complex non-Euclidean geometric properties often found in generalized settings.

Implications and Future Directions

The implications of Knill's results are multifaceted. Theoretically, it reinforces the interconnectedness of graph theory and topological concepts like manifolds and spheres, showcasing how high-dimensional simplicial complexes can exhibit Hamiltonian properties. Practically, it poses significant potential in computational topology and fields like computer graphics, where understanding such cycles aids in efficient mesh traversal and handling complex surface structures.

An open avenue for future research remains the extension of this theory to broader classes of graphs and potential connections to graph coloring problems, akin to the famous Four Color Theorem in two dimensions. Furthermore, exploring the relationships between Hamiltonian properties and other graph invariants like chromatic or edge-connectivity could yield deeper insights into the topological characteristics embedded in graph structures.

Knill's work sits at a fascinating intersection of mathematics and computer science and invites continued exploration into the rich tapestry of graph theory and its applications to complex systems. As the field progresses, leveraging such robust theoretical frameworks will undoubtedly illuminate new pathways in understanding and manipulating complicated combinatorial objects.