Combinatorial manifolds are Hamiltonian

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.