Catalog/Sparsity/AdmBoundByTopGrad/Contract.lean
| 1 | import Catalog.Sparsity.ColoringNumbers.Contract |
| 2 | import Catalog.Sparsity.Admissibility.Contract |
| 3 | import Catalog.Sparsity.ShallowTopologicalMinor.Contract |
| 4 | import Mathlib.Combinatorics.SimpleGraph.Finite |
| 5 | |
| 6 | open Catalog.Sparsity.ColoringNumbers |
| 7 | open Catalog.Sparsity.Admissibility |
| 8 | open Catalog.Sparsity.ShallowTopologicalMinor |
| 9 | |
| 10 | namespace Catalog.Sparsity.AdmBoundByTopGrad |
| 11 | |
| 12 | /-- Lemma 3.2/ch2: the r-admissibility is bounded by the cube of any upper |
| 13 | bound `d` on the edge density of depth-(r-1) topological minors. |
| 14 | |
| 15 | The proof constructs a greedy ordering achieving |
| 16 | adm_r(G,σ) ≤ 1 + 6r · d³. This is equivalent to the source statement |
| 17 | adm_r(G) ≤ 1 + 6r · ⌈∇̃_{r-1}(G)⌉³ when `d = ⌈∇̃_{r-1}(G)⌉`. -/ |
| 18 | axiom adm_le_of_topGrad_bound |
| 19 | {V : Type} [DecidableEq V] [Fintype V] |
| 20 | (G : SimpleGraph V) (r d : ℕ) |
| 21 | (hd : ∀ {W : Type} [DecidableEq W] [Fintype W] |
| 22 | (H : SimpleGraph W) [DecidableRel H.Adj], |
| 23 | IsShallowTopologicalMinor H G (r - 1) → |
| 24 | H.edgeFinset.card ≤ d * Fintype.card W) : |
| 25 | ∃ (ord : LinearOrder V), |
| 26 | letI := ord; adm G r ≤ 1 + 6 * r * d ^ 3 |
| 27 | |
| 28 | end Catalog.Sparsity.AdmBoundByTopGrad |
| 29 |