[#P2804] Flip-graph diameter for triangulations of C(10,4)
Problem. Let C(10,4) be the convex hull in \(\mathbb R^4\) of \((t,t^2,t^3,t^4)\) for \(t=1,\ldots,10\). Form the graph whose vertices are triangulations of this point configuration without added vertices, with two triangulations adjacent when they differ by one bistellar flip. Determine its connected components and the diameter of each component.
1Context
The known finite census makes this a focused reconfiguration problem. Its graph can support later questions about regular triangulations, monotone orientations, and higher Stasheff-Tamari orders.
2Definitions
Definition 1 (A triangulation). A triangulation is a face-to-face subdivision of the polytope into 4-simplices whose vertices are among the ten given points.
Definition 2 (A bistellar flip replaces one triangulation of the convex hull of a circuit by the other triangulation supported on that). A bistellar flip replaces one triangulation of the convex hull of a circuit by the other triangulation supported on that circuit.
Definition 3 (The diameter of a connected graph). The diameter of a connected graph is the largest shortest-path distance between two vertices.
3What counts as a solution
- Enumerate the triangulations with independently checkable canonical hashes, replay all bistellar-flip adjacencies, and certify the component partition and every reported diameter.
1Status
Current status (Current status and unresolved remainder). UNKNOWN as of 2026-08-01: The 2026-08-01 search found a published count of triangulations but no component or diameter result for this exact flip graph.[1]
1Records
Notes and companion material
Original intake status. OPEN as of 2026-08-01. UNKNOWN as of 2026-08-01: The 2026-08-01 search found a published count of triangulations but no component or diameter result for this exact flip graph.
- 2026-07-27: Searches for C(10,4) triangulations found a reverse-search paper reporting 4824 triangulations. Searches pairing the exact polytope with flip-graph diameter returned no value.
- 2026-07-27: The target was checked against earlier candidate files and all live prospecting records with no duplicate.
- A complete artifact should include canonical simplex lists, circuit-supported flip edges, component labels, and shortest-path or eccentricity certificates.
- Fresh exact-title, parameter, source, and corpus searches were completed on 2026-08-01.
Recorded example 1. Pulling the vertices in the order 1 through 10 gives a triangulation and therefore at least one vertex of the flip graph.
Computational notes
- The cyclic 4-polytope has 35 tetrahedral facets by the formula \(n(n-3)/2\). The count of 4824 triangulations is attributed to the published reverse-search comparison and was not independently reproduced.
How the 2 records connect
ProblemFlip-graph diameter for triangulations of C(10,4)
2See also
- Dürer’s edge-unfolding problemcomputational geometry
- Minimum spanning-tree dilation on the regular nonagoncomputational geometry
- Rectilinear crossing number of K_28computational geometry
How to cite
TheoremDB contributors, “Flip-graph diameter for triangulations of C(10,4),” TheoremDB research memory, snapshot of August 1, 2026. https://theoremdb.org/statements/cyclic-polytope-c4-10-flip-diameterThis page as plain text: cyclic-polytope-c4-10-flip-diameter.md
This problem includes 2 records joined by 1 typed links, sourced from doi.org[1], current as of August 1, 2026.
1References
- Packet source. Jörg Rambau, Triangulations of cyclic polytopes and higher Bruhat orders, Mathematika 44 (1997), 162-194. Triangulations and bistellar flips for cyclic polytopes. ↗journal article · primary source · checked 2026-08-01Source use: original summary.This is the primary or maintained source used to check the formulation, neighboring results, and current research boundary.Also cited at Editorial research route recorded 2026-08-01.Source used to assess the problem's recorded status.Source named by the research packet.
- Francisco Santos, The number of triangulations of the cyclic polytope C(n,n-4), 2000. Published reverse-search census containing the 4,824 count for C(10,4). ↗preprint · secondary source · author manuscript PDF checked 2026-08-01 · checked 2026-08-01Source use: original summary.This later or complementary source was checked for equivalent formulations, methods, and possible prior answers.Source used to assess the problem's recorded status.For Flip-graph diameter for triangulations of C(10,4): This later or complementary source was checked for equivalent formulations, methods, and possible prior answers.
Original flip-graph component and diameter target with a finite published triangulation census.