Barnette's conjecture on 4-regular 4-polytopes

Status open medium confidence

Barnette's conjecture that every simple 4-dimensional convex polytope has a Hamiltonian graph remains open. It is attributed to D.W. Barnette and was referenced in Grünbaum (1970). Partial results have confirmed the conjecture for specific families of simple 4-polytopes, including Kalai's squeezed polytopes and their polars (Pfeifle 2001, citing also Hebble–Lee 2000). No complete proof or counterexample has been found.

Cited literature (1)

• partial Kalai's squeezed 3-spheres are polytopal (2001)
  Julian Pfeifle · arXiv preprint · arXiv:math/0110240

  Shows that the polars of Kalai's squeezed polytopes form a new family of simple 4-polytopes supporting Barnette's conjecture that all simple 4-polytopes admit a Hamiltonian circuit; also gives a shorter proof of Hebble–Lee's 2000 result that the dual graphs of these 4-polytopes are Hamiltonian.

Reviewer notes. This is Barnette's lesser-known conjecture (on simple 4-polytopes) and must not be confused with his better-known conjecture on cubic bipartite planar graphs (tracked in the corpus as OPG record 'barnettes_conjecture'). The statement as given — every simple 4-dimensional convex polytope has a Hamiltonian graph — is sometimes called 'Barnette's conjecture on 4-polytopes' or the 'Barnette–Grünbaum conjecture'. The only verified primary source found is Pfeifle (2001, math/0110240), which is pre-2008 and provides partial evidence. WebSearch was unavailable during this review, limiting comprehensiveness; no post-2008 paper claiming a resolution was found across multiple arXiv queries. Confidence is medium rather than high because of the WebSearch outage.

Auto-reviewed 2026-09-10 with claude-sonnet-4-6 (web search enabled).