MathDB

Hamiltonian face-lattice conjecture

For every convex polytope PP of dimension d≥1d\ge 1, let L(P)L(P) be its face lattice and let G(L(P))G(L(P)) be its cover graph: the vertices are the faces of PP, and two faces FF and F′F' are adjacent exactly when one covers the other in L(P)L(P), equivalently when F⊊F′F\subsetneq F' and dim⁡F′=dim⁡F+1\dim F'=\dim F+1 (or conversely). The conjecture asserts that G(L(P))G(L(P)) has a Hamiltonian cycle, i.e. a cycle containing every face of PP exactly once.

References

Primary source

arXiv

Additional references

graph-theorypolyhedral-combinatorics
Posted · edited

Progress summary

Refreshed
Claimed progress

A new paper claims the conjecture for all simplicial polytopes, but the general case remains open.

The conjecture asserts that for every polytope PP of dimension d≥1d \geq 1, the cover graph G(L(P))G(L(P)) of its face lattice has a Hamiltonian cycle. The cycle lists every face once, with consecutive faces related by inclusion and differing in dimension by 11; the supplied record does not identify an original proposer.

Known results

  • A 2024 paper proves the conjecture for hypercubes, permutahedra, BB-permutahedra, associahedra, cyclic polytopes, graph associahedra of chordal graphs, and quotientopes.
  • It also proves the conjecture for all 33-dimensional polytopes and reports no counterexamples.

September 17, 2026 simplicial case

Robert Lauff’s arXiv paper claims Hamiltonicity for the face lattice of every simplicial polytope, using line shellings, cube decompositions, and a Hamilton-cycle theorem. Its abstract explicitly leaves the nonsimplicial case open, so this is substantial claimed progress rather than a complete resolution.

Current status (as of September 2026): the simplicial-polytope case is claimed proved, while the nonsimplicial and hence general cases remain open; the new proof is unverified.

Sources

Solutions 0

No solutions have been posted yet.