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namespace OAI
noncomputable section
namespace ExtremalEdges
open Finset
variable {V I : Type*} [Fintype V] [Fintype I]
attribute [local instance] Classical.propDecidable
local instance capMatchingDecidableEq (carrier : Sort*) : DecidableEq carrier :=
Classical.typeDecidableEq carrier
def edgeEnds (e : I → V × V) (i : I) : Finset V := {(e i).1,(e i).2}
def incident (e : I → V × V) (x : V) : Finset I :=
Finset.univ.filter (fun i => (e i).1 = x ∨ (e i).2 = x)
lemma card_incident (G : SimpleGraph V) [DecidableRel G.Adj] (D : ℕ)
(hd : ∀ x, G.degree x ≤ D) (e : I → V × V)
(he : ∀ i, G.Adj (e i).1 (e i).2) (hi : Function.Injective e) (x : V) :
(incident e x).card ≤ 2*D := by
let L := Finset.univ.filter (fun i => (e i).1 = x)
let R := Finset.univ.filter (fun i => (e i).2 = x)
have hL : L.card ≤ G.degree x := by
rw [← G.card_neighborFinset_eq_degree]
apply Finset.card_le_card_of_injOn (fun i => (e i).2)
· intro i hiL
have hix : (e i).1 = x := (Finset.mem_filter.mp hiL).2
simpa [← hix] using he i
· intro i hiL k hkL hik
apply hi
exact Prod.ext ((Finset.mem_filter.mp hiL).2.trans (Finset.mem_filter.mp hkL).2.symm) hik
have hR : R.card ≤ G.degree x := by
rw [← G.card_neighborFinset_eq_degree]
apply Finset.card_le_card_of_injOn (fun i => (e i).1)
· intro i hiR
have hix : (e i).2 = x := (Finset.mem_filter.mp hiR).2
simpa [← hix] using (he i).symm
· intro i hiR k hkR hik
apply hi
exact Prod.ext hik ((Finset.mem_filter.mp hiR).2.trans (Finset.mem_filter.mp hkR).2.symm)
have heq : incident e x = L ∪ R := by ext i; simp [incident,L,R]
rw [heq]
have hh := Finset.card_union_le L R
have hdx := hd x
omega
theorem large_matching (G : SimpleGraph V) [DecidableRel G.Adj] (D : ℕ)
(hd : ∀ x, G.degree x ≤ D) (e : I → V × V)
(he : ∀ i, G.Adj (e i).1 (e i).2) (hi : Function.Injective e) :
∃ M : Finset I, (M : Set I).Pairwise (fun i k => Disjoint (edgeEnds e i) (edgeEnds e k)) ∧
Fintype.card I ≤ M.card*(4*D) := by
let T : Finset (Finset I) := Finset.univ.filter
(fun M => (M : Set I).Pairwise (fun i k => Disjoint (edgeEnds e i) (edgeEnds e k)))
have hT : T.Nonempty := ⟨∅, by simp [T]⟩
obtain ⟨M,hmax⟩ := T.exists_maximal hT
have hM : (M : Set I).Pairwise (fun i k => Disjoint (edgeEnds e i) (edgeEnds e k)) :=
(Finset.mem_filter.mp hmax.1).2
have cover (i : I) : ∃ m ∈ M, ¬Disjoint (edgeEnds e i) (edgeEnds e m) := by
by_contra hn
push Not at hn
have hiM : i ∉ M := by
intro hh
exact Finset.disjoint_left.mp (hn i hh)
(show (e i).1 ∈ edgeEnds e i by simp [edgeEnds]) (by simp [edgeEnds])
have hpair : ((insert i M : Finset I) : Set I).Pairwise (fun a b => Disjoint (edgeEnds e a) (edgeEnds e b)) := by
intro a ha b hb hab
rcases Finset.mem_insert.mp ha with hai | ham
· subst a
rcases Finset.mem_insert.mp hb with hbi | hbm
· exact False.elim (hab hbi.symm)
· exact hn b hbm
· rcases Finset.mem_insert.mp hb with hbi | hbm
· subst b
exact (hn a ham).symm
· exact hM ham hbm hab
have hmem : insert i M ∈ T := by simp only [T,Finset.mem_filter,Finset.mem_univ,true_and]; exact hpair
exact hiM ((hmax.2 hmem (Finset.subset_insert i M)) (by simp))
let touch (m : I) := incident e (e m).1 ∪ incident e (e m).2
have hall : (Finset.univ : Finset I) ⊆ M.biUnion touch := by
intro i _
obtain ⟨m,hm,hmt⟩ := cover i
refine Finset.mem_biUnion.mpr ⟨m,hm,?_⟩
have hs : ∃ x, x ∈ edgeEnds e i ∧ x ∈ edgeEnds e m := by
simpa only [Finset.not_disjoint_iff] using hmt
rcases hs with ⟨x,hx,hy⟩
simp only [edgeEnds, Finset.mem_insert, Finset.mem_singleton] at hx hy
rcases hx with rfl | rfl <;> rcases hy with hy | hy <;>
simp [touch,incident,hy]
have ht (m : I) : (touch m).card ≤ 4*D := by
have h1 := card_incident G D hd e he hi (e m).1
have h2 := card_incident G D hd e he hi (e m).2
have hh := Finset.card_union_le (incident e (e m).1) (incident e (e m).2)
dsimp [touch]
omega
refine ⟨M,hM,?_⟩
simpa only [Finset.card_univ] using (Finset.card_le_card hall).trans
(Finset.card_biUnion_le_card_mul M touch (4*D) (fun m _ => ht m))
end ExtremalEdges
end
end OAI
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