File size: 2,580 Bytes
a72760f | 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 | import OAI.NumberTheory.Ostmann.Tree.CycleSelection
namespace OAI
namespace Ostmann.Tree
theorem graph_cycle_of_edges_ge_vertices {V : Type*} [Fintype V] [Nonempty V]
(G : SimpleGraph V) [Fintype G.edgeSet]
(hcard : Fintype.card V ≤ Fintype.card G.edgeSet) :
∃ v : V, ∃ p : G.Walk v v, p.IsCycle := by
classical
by_contra! hn
have hacyclic : G.IsAcyclic := fun v p => hn v p
obtain ⟨T,hGT,_,hT⟩ := SimpleGraph.connected_top.exists_isTree_le_of_le_of_isAcyclic
(G := (⊤ : SimpleGraph V)) (H := G) le_top hacyclic
have htcard := hT.card_edgeFinset
have hle := Finset.card_le_card (SimpleGraph.edgeFinset_mono hGT)
rw [SimpleGraph.edgeFinset_card] at hle
omega
def incidenceGraph {E V W : Type*} (left : E → V) (right : E → W) : SimpleGraph (V ⊕ W) where
Adj x y := match x,y with
| .inl v, .inr w => ∃ e, left e=v ∧ right e=w
| .inr w, .inl v => ∃ e, left e=v ∧ right e=w
| _,_ => False
symm := ⟨by rintro (v|w) (v'|w') h <;> exact h⟩
loopless := ⟨by rintro (v|w) h <;> exact h⟩
noncomputable def incidenceEdgeEmbedding {E V W : Type*} (left : E → V) (right : E → W)
(hinj : Function.Injective (fun e => (left e,right e))) :
E ↪ (incidenceGraph left right).edgeSet where
toFun e := ⟨s(Sum.inl (left e),Sum.inr (right e)), ⟨e,rfl,rfl⟩⟩
inj' := by
intro e f h
have he : s(Sum.inl (left e),Sum.inr (right e)) =
s(Sum.inl (left f),Sum.inr (right f)) := congrArg Subtype.val h
apply hinj
rcases Sym2.eq_iff.mp he with he | he
· exact Prod.ext (Sum.inl.inj he.1) (Sum.inr.inj he.2)
· cases he.1
theorem incidence_cycle_or_parallel {E V W : Type*}
[Fintype E] [Fintype V] [Fintype W] [Nonempty E]
(left : E → V) (right : E → W)
(hcard : Fintype.card V+Fintype.card W ≤ Fintype.card E) :
(∃ e f : E, e ≠ f ∧ left e=left f ∧ right e=right f) ∨
∃ v : V ⊕ W, ∃ p : (incidenceGraph left right).Walk v v, p.IsCycle := by
classical
by_cases hinj : Function.Injective (fun e => (left e,right e))
· right
let : Nonempty (V ⊕ W) := ⟨Sum.inl (left (Classical.choice (inferInstance : Nonempty E)))⟩
apply graph_cycle_of_edges_ge_vertices
have hle := Fintype.card_le_of_injective _ (incidenceEdgeEmbedding left right hinj).injective
rw [Fintype.card_sum]
exact hcard.trans hle
· left
unfold Function.Injective at hinj
push Not at hinj
obtain ⟨e,f,he,hef⟩ := hinj
exact ⟨e,f,hef,congrArg Prod.fst he,congrArg Prod.snd he⟩
end Ostmann.Tree
end OAI
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