| import OAI.Combinatorics.Hamiltonian.ExactDual |
|
|
| namespace OAI |
|
|
| /-! Hamiltonian cycles in cubic bipartite three-vertex-connected plane graphs. -/ |
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|
| namespace Barnette.Network |
| noncomputable section |
| open scoped Classical |
| variable {V E F : Type*} [Fintype V] [Fintype E] [Fintype F] |
| [DecidableEq V] (N : Network V E) (D : Network F E) |
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|
| theorem Duality.hamiltonian (h : N.Duality D) (hN : N.Simple) |
| (hc : N.graph.IsRegularOfDegree 3) (h3 : N.ThreeEdgeConnected) |
| (black : V → Prop) (hb : ∀ e, black (N.tail e) ↔ ¬black (N.head e)) : |
| HasHamiltonianCycle N.graph := by |
| obtain ⟨v⟩ := h.connected.nonempty |
| have hs : (N.star v).Nonempty := by |
| apply Finset.card_pos.mp |
| rw [N.card_star hN,hc v] |
| decide |
| obtain ⟨e,_⟩ := hs |
| obtain ⟨P,_,hP,hPc⟩ := h.connected_complement_matching N D hN hc h3 black hb e |
| let H := (N.restrict (Subtype.val : ↥Pᶜ → E)).graph |
| have hdeg : H.IsRegularOfDegree 2 := N.matching_complement_two hN hc P hP |
| obtain ⟨p,hp⟩ := Barnette.SimpleCycle.cycle_at_of_connected_two H hPc hdeg v |
| let f : H →g N.graph := { |
| toFun := id |
| map_rel' := by |
| rintro a b ⟨hne,k,hk⟩ |
| exact ⟨hne,k.val,hk⟩ } |
| exact ⟨v,p.map f,hp.map Function.bijective_id⟩ |
| end |
| end Barnette.Network |
|
|
| namespace Barnette |
| universe u |
| noncomputable section |
| open scoped Classical |
| variable {V : Type u} [Fintype V] {G : SimpleGraph V} |
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|
| lemma ThreeVertexConnected.connected (h : ThreeVertexConnected G) : G.Connected := by |
| let H := G.induce {v | v ∉ (∅ : Finset V)} |
| let f : H →g G := { toFun := Subtype.val, map_rel' := fun h => h } |
| exact (h.2 ∅ (by simp)).map f (fun v => ⟨⟨v,by simp⟩,rfl⟩) |
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|
| theorem main : MainStatement.{u} := by |
| intro V hV dV G dG hc hb hp h3 |
| have hdeq : dV = Classical.decEq V := Subsingleton.elim _ _ |
| subst dV |
| have hdrel : dG = fun a b => Classical.propDecidable (G.Adj a b) := Subsingleton.elim _ _ |
| subst dG |
| obtain ⟨E⟩ := hp |
| have hconn := h3.connected |
| let : Nonempty V := hconn.nonempty |
| have horder := cubic_bipartite_order hc hb |
| obtain ⟨N,F,f,D,hr,hd⟩ := E.exists_exact_dual hconn |
| let : Fintype F := f |
| have hN : N.Simple := by |
| refine ⟨hr.loopless,?_⟩ |
| intro e f hef |
| apply Subtype.ext |
| rw [← hr e,← hr f] |
| exact hef |
| have hc' : N.graph.IsRegularOfDegree 3 := by rw [hr.graph_eq]; exact hc |
| have h3' : ThreeVertexConnected N.graph := hr.graph_eq.symm ▸ h3 |
| have hec : N.ThreeEdgeConnected := by |
| intro A ha ho |
| obtain ⟨w,hw⟩ := ho |
| exact N.cut_card_three h3' (by omega) A ha ⟨w,Finset.mem_compl.mp hw⟩ |
| obtain ⟨color⟩ := hb |
| let black : V → Prop := fun v => color v = 0 |
| have hcolor (e : G.edgeSet) : black (N.tail e) ↔ ¬black (N.head e) := by |
| have hne := color.valid (G.mem_edgeSet.mp ((hr e).symm ▸ e.property)) |
| have ht := (color (N.tail e)).isLt |
| have hh := (color (N.head e)).isLt |
| have hne' : (color (N.tail e)).val ≠ (color (N.head e)).val := fun h => hne (Fin.ext h) |
| change color (N.tail e) = 0 ↔ ¬color (N.head e) = 0 |
| simp only [Fin.ext_iff,Fin.val_zero] |
| omega |
| have hh := hd.hamiltonian N D hN hc' hec black hcolor |
| rwa [hr.graph_eq] at hh |
| end |
| end Barnette |
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| end OAI |
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