| import OAI.Combinatorics.GraphThreshold.GraphContainment |
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| namespace OAI |
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| noncomputable section |
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| open scoped Pointwise |
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| namespace LeanBlast.SecondKahnKalai |
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| variable {n : ℕ} |
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| /-- The subgraph containing exactly the given edges and their incident vertices. |
| For the empty edge set this has no vertices, as required for internal copies. -/ |
| def edgeSubgraph (S : Finset (Edge n)) : (⊤ : SimpleGraph (Fin n)).Subgraph where |
| verts := (graphOfEdges S).support |
| Adj := (graphOfEdges S).Adj |
| adj_sub h := h.ne |
| edge_vert h := ⟨_, h⟩ |
| symm := (graphOfEdges S).symm |
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| def edgeSupportGraph (S : Finset (Edge n)) : |
| SimpleGraph (graphOfEdges S).support := (edgeSubgraph S).coe |
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| @[simp] |
| theorem edgeSubgraph_spanningCoe (S : Finset (Edge n)) : |
| (edgeSubgraph S).spanningCoe = graphOfEdges S := rfl |
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| @[simp] |
| theorem edgeSupportGraph_map (S : Finset (Edge n)) : |
| (edgeSupportGraph S).map (Function.Embedding.subtype |
| (fun v => v ∈ (graphOfEdges S).support)) = graphOfEdges S := by |
| exact SimpleGraph.Subgraph.spanningCoe_coe (edgeSubgraph S) |
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| theorem edgeSupportGraph_noIsolates (S : Finset (Edge n)) |
| (v : (graphOfEdges S).support) : |
| ∃ w, (edgeSupportGraph S).Adj v w := by |
| obtain ⟨w, hw⟩ := v.property |
| have hvw : (graphOfEdges S).Adj v w := hw |
| exact ⟨⟨w, ⟨v, hvw.symm⟩⟩, hvw⟩ |
|
|
| /-- Restriction of a graph isomorphism to its incident vertices. -/ |
| def supportIso {V W : Type*} {G : SimpleGraph V} {H : SimpleGraph W} |
| (e : G ≃g H) : G.induce G.support ≃g H.induce H.support where |
| toFun v := ⟨e v, by |
| obtain ⟨w, hw⟩ := v.property |
| exact ⟨e w, e.toHom.map_adj hw⟩⟩ |
| invFun v := ⟨e.symm v, by |
| obtain ⟨w, hw⟩ := v.property |
| exact ⟨e.symm w, e.symm.toHom.map_adj hw⟩⟩ |
| left_inv v := Subtype.ext (e.symm_apply_apply v) |
| right_inv v := Subtype.ext (e.apply_symm_apply v) |
| map_rel_iff' := e.map_rel_iff |
|
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| def edgeSupportGraph_smulIso (σ : Equiv.Perm (Fin n)) (S : Finset (Edge n)) : |
| edgeSupportGraph S ≃g edgeSupportGraph (σ • S) := by |
| have e : graphOfEdges S ≃g graphOfEdges (σ • S) := by |
| rw [graphOfEdges_smul] |
| exact SimpleGraph.Iso.map σ _ |
| exact supportIso e |
|
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| theorem map_eq_of_iso {V W : Type*} {G : SimpleGraph V} {H : SimpleGraph W} |
| (e : G ≃g H) : G.map e.toEquiv.toEmbedding = H := by |
| ext a b |
| rw [SimpleGraph.map_adj] |
| constructor |
| · rintro ⟨v, w, hvw, rfl, rfl⟩ |
| exact e.toHom.map_adj hvw |
| · intro hab |
| exact ⟨e.symm a, e.symm b, e.symm.toHom.map_adj hab, |
| e.apply_symm_apply a, e.apply_symm_apply b⟩ |
|
|
| /-- An isomorphism of incident-vertex graphs extends to a permutation of the |
| full ambient vertex set, hence to equality of the corresponding edge copies. -/ |
| theorem exists_smul_eq_of_edgeSupportGraph_iso {S T : Finset (Edge n)} |
| (e : edgeSupportGraph S ≃g edgeSupportGraph T) : |
| ∃ σ : Equiv.Perm (Fin n), σ • S = T := by |
| let f : (graphOfEdges S).support ↪ Fin n := Function.Embedding.subtype _ |
| let j : (graphOfEdges T).support ↪ Fin n := Function.Embedding.subtype _ |
| let g : (graphOfEdges S).support ↪ Fin n := e.toEquiv.toEmbedding.trans j |
| obtain ⟨σ, hσ⟩ := exists_perm_comp_embedding f g |
| have hfun : (fun v => σ (f v)) = (fun v => j (e v)) := funext hσ |
| have hm : ((edgeSupportGraph S).map f).map σ.toEmbedding = |
| (edgeSupportGraph T).map j := by |
| rw [SimpleGraph.map_map] |
| change (edgeSupportGraph S).map (fun v => σ (f v)) = _ |
| rw [hfun] |
| change (edgeSupportGraph S).map (j ∘ e.toEquiv.toEmbedding) = _ |
| rw [← SimpleGraph.map_map, map_eq_of_iso] |
| refine ⟨σ, (graphEdgeEquiv n).injective ?_⟩ |
| change graphOfEdges (σ • S) = graphOfEdges T |
| rw [graphOfEdges_smul] |
| simpa only [f, j, edgeSupportGraph_map] using hm |
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|
| theorem subgraph_eq_edgeSubgraph |
| (K : (⊤ : SimpleGraph (Fin n)).Subgraph) |
| (hK : ∀ v : K.verts, ∃ w, K.coe.Adj v w) : |
| K = edgeSubgraph (graphEdges K.spanningCoe) := by |
| apply SimpleGraph.Subgraph.ext |
| · change K.verts = (graphOfEdges (graphEdges K.spanningCoe)).support |
| rw [graphOfEdges_graphEdges] |
| ext v |
| constructor |
| · intro hv |
| obtain ⟨w, hw⟩ := hK ⟨v, hv⟩ |
| exact ⟨w, hw⟩ |
| · rintro ⟨w, hw⟩ |
| exact K.edge_vert hw |
| · change K.Adj = (graphOfEdges (graphEdges K.spanningCoe)).Adj |
| rw [graphOfEdges_graphEdges] |
| rfl |
|
|
| theorem subgraph_eq_edgeSubgraph_of_iso {S : Finset (Edge n)} |
| (K : (⊤ : SimpleGraph (Fin n)).Subgraph) |
| (e : edgeSupportGraph S ≃g K.coe) : |
| K = edgeSubgraph (graphEdges K.spanningCoe) := by |
| apply subgraph_eq_edgeSubgraph K |
| intro v |
| obtain ⟨w, hw⟩ := edgeSupportGraph_noIsolates S (e.symm v) |
| refine ⟨e w, ?_⟩ |
| have hh : K.coe.Adj (e (e.symm v)) (e w) := e.toHom.map_adj hw |
| simpa only [e.apply_symm_apply] using hh |
|
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| theorem edgeCount_edgeSupportGraph (S : Finset (Edge n)) : |
| edgeCount (edgeSupportGraph S) = S.card := by |
| classical |
| have h := card_mappedGraphEdges |
| (Function.Embedding.subtype (fun v => v ∈ (graphOfEdges S).support)) |
| (edgeSupportGraph S) |
| simpa only [edgeSupportGraph_map, graphEdges_graphOfEdges] using h.symm |
|
|
| /-- A selected edge subset of an embedded graph pulls back to an actual copy |
| inside the original graph, using only incident vertices of the selection. -/ |
| def edgeSupportGraph_copy_of_subset {V : Type*} (f : V ↪ Fin n) |
| (H : SimpleGraph V) (S : Finset (Edge n)) |
| (hS : S ⊆ graphEdges (H.map f)) : (edgeSupportGraph S).Copy H := by |
| have hle : graphOfEdges S ≤ H.map f := by |
| simpa only [graphOfEdges_graphEdges] using graphOfEdges_mono hS |
| have hrange (v : (graphOfEdges S).support) : ∃ a : V, f a = v.val := by |
| obtain ⟨w, hw⟩ := v.property |
| obtain ⟨a, b, hab, ha, hb⟩ := (SimpleGraph.map_adj f H v w).mp (hle hw) |
| exact ⟨a, ha⟩ |
| let g : (graphOfEdges S).support → V := fun v => Classical.choose (hrange v) |
| have hg (v : (graphOfEdges S).support) : f (g v) = v.val := |
| Classical.choose_spec (hrange v) |
| refine { toHom := ⟨g, ?_⟩, injective' := ?_ } |
| · intro v w hvw |
| apply (SimpleGraph.map_adj_apply (f := f)).mp |
| rw [hg v, hg w] |
| exact hle hvw |
| · intro v w hvw |
| change g v = g w at hvw |
| apply Subtype.ext |
| rw [← hg v, ← hg w, hvw] |
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| theorem exists_subgraph_iso_edgeSupportGraph {V : Type*} (f : V ↪ Fin n) |
| (H : SimpleGraph V) (S : Finset (Edge n)) |
| (hS : S ⊆ graphEdges (H.map f)) : |
| ∃ F : H.Subgraph, Nonempty (edgeSupportGraph S ≃g F.coe) := |
| SimpleGraph.isContained_iff_exists_iso_subgraph.mp |
| ⟨edgeSupportGraph_copy_of_subset f H S hS⟩ |
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| end LeanBlast.SecondKahnKalai |
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| end |
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| end OAI |
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