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e82fac3 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 | import OAI.Combinatorics.PortMatching.Basic
import Mathlib.Combinatorics.SimpleGraph.DegreeSum
namespace OAI
/-! Perfect-matching-preserving port-path degree reduction for finite simple graphs. -/
noncomputable section
namespace Matching205
universe u
variable {V : Type u} [Fintype V] (G : SimpleGraph V) [DecidableRel G.Adj]
noncomputable instance graphAdjDecidable : DecidableRel (graph G).Adj :=
fun _ _ => Classical.propDecidable _
private lemma size_vertex_ext {x y : Vertex G}
(hbase : x.1 = y.1) (hpos : x.2.val = y.2.val) : x = y := by
rcases x with ⟨v, i⟩
rcases y with ⟨w, j⟩
dsimp at hbase hpos
subst w
congr 1
exact Fin.ext hpos
private lemma size_port_injective (v : V) : Function.Injective (port G v) := by
intro a b hab
have h := congrArg (fun x : Vertex G => x.2.val) hab
change 2 * ((Fintype.equivFin (G.neighborSet v)) a).val =
2 * ((Fintype.equivFin (G.neighborSet v)) b).val at h
apply (Fintype.equivFin (G.neighborSet v)).injective
apply Fin.ext
omega
private lemma size_external_base_ne {x y : Vertex G} (h : External G x y) :
x.1 ≠ y.1 := by
obtain ⟨h, _, _⟩ := h
exact h.ne
private lemma size_external_unique {x y z : Vertex G}
(hy : External G x y) (hz : External G x z) : y = z := by
obtain ⟨hxy, hx, hy⟩ := hy
obtain ⟨hxz, hx', hz⟩ := hz
have hn := size_port_injective G x.1 (hx.symm.trans hx')
have hbase : y.1 = z.1 := congrArg Subtype.val hn
have transport (v w : V) (h : v = w) (hv : G.Adj v x.1) (hw : G.Adj w x.1) :
port G v ⟨x.1, hv⟩ = port G w ⟨x.1, hw⟩ := by
subst w
rfl
have hp := transport y.1 z.1 hbase hxy.symm hxz.symm
exact hy.trans (hp.trans hz.symm)
private def size_neighborCode (x : Vertex G) (y : (graph G).neighborSet x) : Fin 3 := by
classical
exact if y.val.1 = x.1 then
if y.val.2.val + 1 = x.2.val then 0 else 1
else 2
private lemma size_neighborCode_injective (x : Vertex G) :
Function.Injective (size_neighborCode G x) := by
classical
intro y z hc
have hy : Internal G x y.val ∨ External G x y.val := y.property
have hz : Internal G x z.val ∨ External G x z.val := z.property
apply Subtype.ext
rcases hy with hy | hy <;> rcases hz with hz | hz
· apply size_vertex_ext G (hy.1.symm.trans hz.1)
have hv := congrArg Fin.val hc
by_cases hyp : y.val.2.val + 1 = x.2.val <;>
by_cases hzp : z.val.2.val + 1 = x.2.val
all_goals simp [size_neighborCode, hy.1.symm, hz.1.symm, hyp, hzp] at hv
all_goals rcases hy.2 with hy' | hy' <;> rcases hz.2 with hz' | hz' <;> omega
· have hn : z.val.1 ≠ x.1 := (size_external_base_ne G hz).symm
have hv := congrArg Fin.val hc
by_cases hp : y.val.2.val + 1 = x.2.val <;>
simp [size_neighborCode, hy.1.symm, hn, hp] at hv
· have hn : y.val.1 ≠ x.1 := (size_external_base_ne G hy).symm
have hv := congrArg Fin.val hc
by_cases hp : z.val.2.val + 1 = x.2.val <;>
simp [size_neighborCode, hz.1.symm, hn, hp] at hv
· exact size_external_unique G hy hz
theorem graph_degree_le_three (x : Vertex G) : (graph G).degree x ≤ 3 := by
classical
rw [← SimpleGraph.card_neighborSet_eq_degree]
simpa using Fintype.card_le_of_injective (size_neighborCode G x)
(size_neighborCode_injective G x)
theorem card_vertex_add_card (hpos : ∀ v : V, 0 < G.degree v) :
Fintype.card (Vertex G) + Fintype.card V = 4 * G.edgeFinset.card := by
classical
have hfiber (v : V) : Fintype.card (Fin (2 * arity G v - 1)) + 1 =
2 * G.degree v := by
rw [card_fiber G v]
have hv := hpos v
omega
calc
Fintype.card (Vertex G) + Fintype.card V =
∑ v : V, (Fintype.card (Fin (2 * arity G v - 1)) + 1) := by
simp [Vertex, Fintype.card_sigma, Finset.sum_add_distrib]
_ = ∑ v : V, 2 * G.degree v := Finset.sum_congr rfl (fun v _ => hfiber v)
_ = 2 * ∑ v : V, G.degree v := (Finset.mul_sum _ _ _).symm
_ = 4 * G.edgeFinset.card := by
rw [G.sum_degrees_eq_twice_card_edges]
omega
theorem card_vertex (hpos : ∀ v : V, 0 < G.degree v) :
Fintype.card (Vertex G) = 4 * G.edgeFinset.card - Fintype.card V := by
have h := card_vertex_add_card G hpos
omega
end Matching205
end
end OAI
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