| import Mathlib.Combinatorics.SimpleGraph.Coloring.Vertex |
| import Mathlib.Algebra.BigOperators.Group.Finset.Basic |
| import Mathlib.Algebra.Order.BigOperators.Group.Finset |
|
|
| namespace OAI |
|
|
| /-! A chromatic lower bound for finite graphs with independence number at most two. -/ |
|
|
| namespace HadwigerMatching |
|
|
| open scoped BigOperators |
|
|
| variable {V : Type*} [Fintype V] (G : SimpleGraph V) |
|
|
| /-- Every color fiber is an independent vertex set, so a coloring by `k` |
| colors can cover at most `2*k` vertices under the stated independence bound. -/ |
| theorem card_le_twice_of_coloring {k : ℕ} (C : G.Coloring (Fin k)) |
| (hα : ∀ S : Finset V, G.IsIndepSet (S : Set V) → S.card ≤ 2) : |
| Fintype.card V ≤ 2 * k := by |
| classical |
| let fiber : Fin k → Finset V := fun c => Finset.univ.filter (fun v => C v = c) |
| have hfiber (c : Fin k) : (fiber c).card ≤ 2 := by |
| apply hα |
| intro v hv w hw _ hadj |
| exact C.valid hadj ((Finset.mem_filter.mp hv).2.trans |
| (Finset.mem_filter.mp hw).2.symm) |
| have hcount : ∑ c : Fin k, (fiber c).card = Fintype.card V := by |
| simp only [fiber, Finset.card_eq_sum_ones, Finset.sum_filter] |
| rw [Finset.sum_comm] |
| simp |
| calc |
| Fintype.card V = ∑ c : Fin k, (fiber c).card := hcount.symm |
| _ ≤ ∑ _c : Fin k, 2 := Finset.sum_le_sum (fun c _ => hfiber c) |
| _ = 2 * k := by simp [Nat.mul_comm] |
|
|
| /-- A finite graph has finite chromatic number; an optimal coloring |
| then proves the lower bound, including the empty vertex type. -/ |
| theorem card_le_twice_chromaticNumber |
| (hα : ∀ S : Finset V, G.IsIndepSet (S : Set V) → S.card ≤ 2) : |
| Fintype.card V ≤ 2 * G.chromaticNumber.toNat := by |
| have hfinite : G.chromaticNumber ≠ ⊤ := by |
| intro htop |
| have h := G.chromaticNumber_le_card |
| rw [htop] at h |
| simp at h |
| obtain ⟨C⟩ := G.colorable_of_chromaticNumber_ne_top hfinite |
| exact card_le_twice_of_coloring G C hα |
|
|
| /-- A graph with independence number at most two needs at least half as many |
| colors as vertices. -/ |
| theorem card_le_twice_chromaticNumber_of_indepNum (hα : G.indepNum ≤ 2) : |
| Fintype.card V ≤ 2 * G.chromaticNumber.toNat := by |
| apply card_le_twice_chromaticNumber G |
| intro S hS |
| exact hS.card_le_indepNum.trans hα |
|
|
| end HadwigerMatching |
|
|
| end OAI |
|
|
