Hamiltonicity in (0–1)-polyhedra

The graph G(P) of a polyhedron P has a node corresponding to each vertex of P and two nodes are adjacent in G(P) if and only if the corresponding vertices of P are adjacent on P. We show that if P ? $\text{R}$ⁿ is a polyhedron, all of whose vertices have (0–1)-valued coordinates, then (i) if G(P) is bipartite, the G(P) is a hypercube; (ii) if G(P) is nonbipartite, then G(P) is hamilton connected. It is shown that if P ? $\text{R}$ⁿ has (0–1)-valued vertices and is of dimension d (≤n) then there exists a polyhedron P′ ? $\text{R}$^d having (0–1)-valued vertices such that $\text{G(P) ? G(P′)}$. Some combinatorial consequences of these results are also discussed.