Triangular dart data gives transitive local rotation on three or four vertices
ProvedWilliamsonShmoys.three_or_four_vertex_triangular_rotation_transitive__p2m_b6f5758e_dc62_475d_9052_43aa820ea959_a2approximation-algorithmsplanar-graphstreewidth
When the complete graph has three or four vertices, the supplied bijective triangular dart data makes every fixed-tail dart set a single cycle under the induced successor.
Preamble
import Definitions.Def_WilliamsonShmoys_TriangularMap import Theorems.Thm_WilliamsonShmoys_three_vertex_triangular_rotation_transitive__p2m_b6f5758e_dc62_475d_9052_43aa820ea959_a1 import Mathlib set_option autoImplicit false open WilliamsonShmoys
Formal statement
theorem WilliamsonShmoys.three_or_four_vertex_triangular_rotation_transitive__p2m_b6f5758e_dc62_475d_9052_43aa820ea959_a2
{V : Type} [Fintype V] [LinearOrder V]
(H : SimpleGraph V)
(hVlo : 3 ≤ Fintype.card V)
(hVhi : Fintype.card V ≤ 4)
(F : Type) [Fintype F]
(corner : F → Fin 3 → V)
(dart : (F × Fin 3) ≃ H.Dart)
(hTail : ∀ f i, (dart (f, i)).fst = corner f i)
(hHead : ∀ f i, (dart (f, i)).snd = corner f (i + 1))
(hInjective : ∀ f, Function.Injective (corner f))
(hCovered : ∀ v, ∃ f i, corner f i = v)
(hcomplete : ∀ u v : V, u ≠ v → H.Adj u v) :
∀ d e : H.Dart, d.fst = e.fst →
∃ k : Nat,
(fun a : H.Dart =>
dart ((dart.symm a.symm).1, (dart.symm a.symm).2 + 1))^[k] d = e := by sorrySource
David P. Williamson and David B. Shmoys, The Design of Approximation Algorithms, Cambridge University Press, 2011, author electronic manuscript, Section 10.2, proof of Theorem 10.11, pp. 270-272. https://doi.org/10.1017/CBO9780511921735. Bounded tree-width of the pieces: Robertson–Seymour (planar graphs of radius r have tree-width at most 3r+1) / Bodlaender (k-outerplanar graphs have tree-width at most 3k-1).