| 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 | |