feat(combinatorics/simple_graph): Locally linear graphs

src/combinatorics/simple_graph/triangle/basic.lean

@@ -3,7 +3,10 @@ Copyright (c) 2022 Yaël Dillies, Bhavik Mehta. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yaël Dillies, Bhavik Mehta
 -/
+import combinatorics.double_counting
 import combinatorics.simple_graph.clique
+import data.nat.parity
+import data.sym.card
 
 /-!
 # Triangles in graphs
@@ -18,6 +21,8 @@ This module defines and proves properties about triangles in simple graphs.
 
 ## Main declarations
 
+* `simple_graph.edge_disjoint_triangles`: Predicate for a graph whose triangles are edge-disjoint.
+* `simple_graph.locally_linear`: Predicate for each edge in a graph to be in a unique triangle.
 * `simple_graph.far_from_triangle_free`: Predicate for a graph to have enough triangles that, to
   remove all of them, one must one must remove a lot of edges. This is the crux of the Triangle
   Removal lemma.
@@ -28,21 +33,183 @@ This module defines and proves properties about triangles in simple graphs.
 * Find a better name for `far_from_triangle_free`. Added 4/26/2022. Remove this TODO if it gets old.
 -/
 
-open finset fintype nat
-open_locale classical
+namespace finset
+variables {α : Type*} [decidable_eq α]
+
+lemma insert_comm (a b : α) (s : finset α) : insert a (insert b s) = insert b (insert a s) :=
+coe_injective $ by { push_cast, exact set.insert_comm _ _ _ }
+
+end finset
+
+namespace simple_graph
+variables {V : Type*}
+
+@[simp] lemma edge_set_top : (⊤ : simple_graph V).edge_set = {e : sym2 V | ¬ e.is_diag} :=
+by ext x; induction x using sym2.ind; simp
+
+@[simp] lemma edge_finset_top [fintype V] [decidable_eq V] [fintype (⊤ : simple_graph V).edge_set] :
+  (⊤ : simple_graph V).edge_finset = finset.univ.filter (λ e, ¬ e.is_diag) :=
+by ext x; induction x using sym2.ind; simp
+
+end simple_graph
+
+open finset fintype (card) nat
 
 namespace simple_graph
-variables {α 𝕜 : Type*} [fintype α] [linear_ordered_field 𝕜] {G H : simple_graph α} {ε δ : 𝕜}
-  {n : ℕ} {s : finset α}
+variables {α β 𝕜 : Type*} [linear_ordered_field 𝕜] {G H : simple_graph α} {ε δ : 𝕜} {n : ℕ}
+  {s : finset α}
+
+section locally_linear
+
+/-- A graph has edge-disjoint triangles if each edge belongs to at most one triangle. -/
+def edge_disjoint_triangles (G : simple_graph α) : Prop :=
+(G.clique_set 3).pairwise $ λ x y, (x ∩ y : set α).subsingleton
+
+/-- A graph is locally linear if each edge belongs to exactly one triangle. -/
+def locally_linear (G : simple_graph α) : Prop :=
+G.edge_disjoint_triangles ∧ ∀ ⦃x y⦄, G.adj x y → ∃ s, G.is_n_clique 3 s ∧ x ∈ s ∧ y ∈ s
+
+protected lemma locally_linear.edge_disjoint_triangles :
+  G.locally_linear → G.edge_disjoint_triangles :=
+and.left
+
+lemma edge_disjoint_triangles.mono (h : G ≤ H) (hH : H.edge_disjoint_triangles) :
+  G.edge_disjoint_triangles :=
+hH.mono $ clique_set_mono h
+
+@[simp] lemma edge_disjoint_triangles_bot : (⊥ : simple_graph α).edge_disjoint_triangles :=
+by simp [edge_disjoint_triangles]
+
+@[simp] lemma locally_linear_bot : (⊥ : simple_graph α).locally_linear := by simp [locally_linear]
+
+lemma edge_disjoint_triangles.map (f : α ↪ β) (hG : G.edge_disjoint_triangles) :
+  (G.map f).edge_disjoint_triangles :=
+begin
+  rw [edge_disjoint_triangles, clique_set_map (bit1_lt_bit1.2 zero_lt_one),
+    ((finset.map_injective f).inj_on _).pairwise_image],
+  classical,
+  rintro s hs t ht hst,
+  dsimp [function.on_fun],
+  rw [←coe_inter, ←map_inter, coe_map, coe_inter],
+  exact (hG hs ht hst).image _,
+end
+
+lemma locally_linear.map (f : α ↪ β) (hG : G.locally_linear) : (G.map f).locally_linear :=
+begin
+  refine ⟨hG.1.map _, _⟩,
+  rintro _ _ ⟨a, b, h, rfl, rfl⟩,
+  obtain ⟨s, hs, ha, hb⟩ := hG.2 h,
+  exact ⟨s.map f, hs.map, mem_map_of_mem _ ha, mem_map_of_mem _ hb⟩,
+end
+
+@[simp] lemma locally_linear_comap {G : simple_graph β} {e : α ≃ β} :
+  (G.comap e).locally_linear ↔ G.locally_linear :=
+begin
+  refine ⟨λ h, _, _⟩,
+  { rw [←comap_map_eq e.symm.to_embedding G, comap_symm, map_symm],
+    exact h.map _ },
+  { rw [←equiv.coe_to_embedding, ←map_symm],
+    exact locally_linear.map _ }
+end
+
+variables [decidable_eq α]
+
+lemma edge_disjoint_triangles_iff_mem_sym2_subsingleton :
+  G.edge_disjoint_triangles ↔
+    ∀ ⦃e : sym2 α⦄, ¬e.is_diag → {s ∈ G.clique_set 3 | e ∈ (s : finset α).sym2}.subsingleton :=
+begin
+  have : ∀ a b, a ≠ b → {s ∈ (G.clique_set 3 : set (finset α)) | ⟦(a, b)⟧ ∈ (s : finset α).sym2}
+    = {s | G.adj a b ∧ ∃ c, G.adj a c ∧ G.adj b c ∧ s = {a, b, c}},
+  { rintro a b hab,
+    ext s,
+    simp only [mem_sym2_iff, sym2.mem_iff, forall_eq_or_imp, forall_eq, set.sep_and,
+      set.mem_inter_iff, set.mem_sep_iff, mem_clique_set_iff, set.mem_set_of_eq,
+      and_and_and_comm _ (_ ∈ _), and_self, is_3_clique_iff],
+    split,
+    { rintro ⟨⟨c, d, e, hcd, hce, hde, rfl⟩, hab⟩,
+      simp only [mem_insert, mem_singleton] at hab,
+      obtain ⟨rfl | rfl | rfl, rfl | rfl | rfl⟩ := hab;
+      simp only [*, adj_comm, true_and, ne.def, eq_self_iff_true, not_true] at *;
+      refine ⟨c, _⟩ <|> refine ⟨d, _⟩ <|> refine ⟨e, _⟩; simp [*, pair_comm, insert_comm] },
+    { rintro ⟨hab, c, hac, hbc, rfl⟩,
+      refine ⟨⟨a, b, c, _⟩, _⟩; simp [*] } },
+  split,
+  { rw sym2.forall,
+    rintro hG a b hab,
+    simp only [sym2.is_diag_iff_proj_eq] at hab,
+    rw this _ _ (sym2.mk_is_diag_iff.not.2 hab),
+    rintro _ ⟨hab, c, hac, hbc, rfl⟩ _ ⟨-, d, had, hbd, rfl⟩,
+    refine hG.eq _ _ (set.nontrivial.not_subsingleton ⟨a, _, b, _, hab.ne⟩);
+    simp [is_3_clique_triple_iff, *] },
+  { simp only [edge_disjoint_triangles, is_3_clique_iff, set.pairwise, mem_clique_set_iff, ne.def,
+      forall_exists_index, and_imp, ←@set.not_nontrivial_iff _ (_ ∩ _), not_imp_not, set.nontrivial,
+      set.mem_inter_iff, mem_coe],
+    rintro hG _ a b c hab hac hbc rfl _ d e f hde hdf hef rfl g hg₁ hg₂ h hh₁ hh₂ hgh,
+    refine hG (sym2.mk_is_diag_iff.not.2 hgh)_ _; simp [is_3_clique_triple_iff, *] }
+end
+
+alias edge_disjoint_triangles_iff_mem_sym2_subsingleton ↔
+  edge_disjoint_triangles.mem_sym2_subsingleton _
+
+variables [fintype α] [decidable_rel G.adj]
+
+instance : decidable G.edge_disjoint_triangles :=
+decidable_of_iff ((G.clique_finset 3 : set (finset α)).pairwise $ λ x y, (x ∩ y).card ≤ 1) $
+  by simpa only [coe_clique_finset, edge_disjoint_triangles, finset.card_le_one, ←coe_inter]
+
+instance : decidable G.locally_linear := and.decidable
+
+lemma edge_disjoint_triangles.card_edge_finset_le (hG : G.edge_disjoint_triangles) :
+  3 * (G.clique_finset 3).card ≤ G.edge_finset.card :=
+begin
+  rw [mul_comm, ←mul_one G.edge_finset.card],
+  refine card_mul_le_card_mul (λ s e, e ∈ s.sym2) _ (λ e he, _),
+  { simp only [is_3_clique_iff, mem_clique_finset_iff, mem_sym2_iff,
+      forall_exists_index, and_imp],
+    rintro _ a b c hab hac hbc rfl,
+    have : finset.card ({⟦(a, b)⟧, ⟦(a, c)⟧, ⟦(b, c)⟧} : finset $ sym2 α) = 3,
+    { refine card_eq_three.2 ⟨_, _, _, _, _, _, rfl⟩; simp [hab.ne, hac.ne, hbc.ne] },
+    rw ←this,
+    refine card_mono _,
+    simp [insert_subset, *] },
+  { simpa only [card_le_one, mem_bipartite_below, and_imp, set.subsingleton, set.mem_sep_iff,
+      mem_clique_finset_iff, mem_clique_set_iff]
+      using hG.mem_sym2_subsingleton (G.not_is_diag_of_mem_edge_set $ mem_edge_finset.1 he) }
+end
+
+lemma locally_linear.card_edge_finset (hG : G.locally_linear) :
+  G.edge_finset.card = 3 * (G.clique_finset 3).card :=
+begin
+  refine hG.edge_disjoint_triangles.card_edge_finset_le.antisymm' _,
+  rw [←mul_comm, ←mul_one (finset.card _)],
+  refine card_mul_le_card_mul (λ e s, e ∈ s.sym2) _ _,
+  { simpa [sym2.forall, nat.succ_le_iff, card_pos, finset.nonempty] using hG.2 },
+  simp only [mem_clique_finset_iff, is_3_clique_iff, forall_exists_index, and_imp],
+  rintro _ a b c hab hac hbc rfl,
+  refine (card_mono _).trans _,
+  { exact {⟦(a, b)⟧, ⟦(a, c)⟧, ⟦(b, c)⟧} },
+  { simp only [subset_iff, sym2.forall, mem_sym2_iff, le_eq_subset, mem_bipartite_below, mem_insert,
+      mem_edge_finset, mem_singleton, and_imp, mem_edge_set, sym2.mem_iff, forall_eq_or_imp,
+      forall_eq, quotient.eq, sym2.rel_iff],
+    rintro d e hde (rfl | rfl | rfl) (rfl | rfl | rfl); simp [*] at * },
+  { exact (card_insert_le _ _).trans (succ_le_succ $ (card_insert_le _ _).trans_eq $
+      by rw card_singleton) }
+end
+
+end locally_linear
+
+variables (G ε) [fintype α] [decidable_eq α] [decidable_rel G.adj] [decidable_rel H.adj]
 
 /-- A simple graph is *`ε`-triangle-free far* if one must remove at least `ε * (card α)^2` edges to
 make it triangle-free. -/
-def far_from_triangle_free (G : simple_graph α) (ε : 𝕜) : Prop :=
-G.delete_far (λ H, H.clique_free 3) $ ε * (card α^2 : ℕ)
+def far_from_triangle_free : Prop := G.delete_far (λ H, H.clique_free 3) $ ε * (card α^2 : ℕ)
+
+variables {G ε}
 
 lemma far_from_triangle_free_iff :
   G.far_from_triangle_free ε ↔
-    ∀ ⦃H⦄, H ≤ G → H.clique_free 3 → ε * (card α^2 : ℕ) ≤ G.edge_finset.card - H.edge_finset.card :=
+    ∀ ⦃H : simple_graph _⦄ [decidable_rel H.adj], by exactI
+      H ≤ G → H.clique_free 3 → ε * (card α^2 : ℕ) ≤ G.edge_finset.card - H.edge_finset.card :=
 delete_far_iff
 
 alias far_from_triangle_free_iff ↔ far_from_triangle_free.le_card_sub_card _
@@ -51,14 +218,87 @@ lemma far_from_triangle_free.mono (hε : G.far_from_triangle_free ε) (h : δ 
   G.far_from_triangle_free δ :=
 hε.mono $ mul_le_mul_of_nonneg_right h $ cast_nonneg _
 
+@[simp] lemma far_from_triangle_free_of_nonpos (hε : ε ≤ 0) : G.far_from_triangle_free ε :=
+λ H _ _, (mul_nonpos_of_nonpos_of_nonneg hε $ nat.cast_nonneg _).trans $ nat.cast_nonneg _
+
 lemma far_from_triangle_free.clique_finset_nonempty' (hH : H ≤ G) (hG : G.far_from_triangle_free ε)
   (hcard : (G.edge_finset.card - H.edge_finset.card : 𝕜) < ε * (card α ^ 2 : ℕ)) :
   (H.clique_finset 3).nonempty :=
 nonempty_of_ne_empty $ H.clique_finset_eq_empty_iff.not.2 $ λ hH',
   (hG.le_card_sub_card hH hH').not_lt hcard
 
+private lemma far_from_triangle_free_of_disjoint_triangles_aux {tris : finset (finset α)}
+  (htris : tris ⊆ G.clique_finset 3)
+  (pd : (tris : set (finset α)).pairwise (λ x y, (x ∩ y : set α).subsingleton)) (hHG : H ≤ G)
+  (hH : H.clique_free 3) : tris.card ≤ G.edge_finset.card - H.edge_finset.card :=
+begin
+  rw [←card_sdiff (edge_finset_mono hHG), ←card_attach],
+  by_contra' hG,
+  have : ∀ ⦃t⦄, t ∈ tris → ∃ x y, x ∈ t ∧ y ∈ t ∧ x ≠ y ∧ ⟦(x, y)⟧ ∈ G.edge_finset \ H.edge_finset,
+  { intros t ht,
+    by_contra' h,
+    refine hH t _,
+    simp only [not_and, mem_sdiff, not_not, mem_edge_finset, mem_edge_set] at h,
+    obtain ⟨x, y, z, xy, xz, yz, rfl⟩ := is_3_clique_iff.1 (G.mem_clique_finset_iff.1 $ htris ht),
+    rw is_3_clique_triple_iff,
+    refine ⟨h _ _ _ _ xy.ne xy, h _ _ _ _ xz.ne xz, h _ _ _ _ yz.ne yz⟩; simp },
+  choose fx fy hfx hfy hfne fmem using this,
+  let f : {x // x ∈ tris} → sym2 α := λ t, ⟦(fx t.2, fy t.2)⟧,
+  have hf : ∀ x, x ∈ tris.attach → f x ∈ G.edge_finset \ H.edge_finset := λ x hx, fmem _,
+  obtain ⟨⟨t₁, ht₁⟩, -, ⟨t₂, ht₂⟩, -, tne, t : ⟦_⟧ = ⟦_⟧⟩ :=
+    exists_ne_map_eq_of_card_lt_of_maps_to hG hf,
+  dsimp at t,
+  have i := pd ht₁ ht₂ (subtype.val_injective.ne tne),
+  rw sym2.eq_iff at t,
+  cases t,
+  { exact hfne _ (i ⟨hfx ht₁, t.1.symm ▸ hfx ht₂⟩ ⟨hfy ht₁, t.2.symm ▸ hfy ht₂⟩) },
+  { exact hfne _ (i ⟨hfx ht₁, t.1.symm ▸ hfy ht₂⟩ ⟨hfy ht₁, t.2.symm ▸ hfx ht₂⟩) }
+end
+
+/-- If there are `ε * (card α)^2` disjoint triangles, then the graph is `ε`-far from being
+triangle-free. -/
+lemma far_from_triangle_free_of_disjoint_triangles (tris : finset (finset α))
+  (htris : tris ⊆ G.clique_finset 3)
+  (pd : (tris : set (finset α)).pairwise (λ x y, (x ∩ y : set α).subsingleton))
+  (tris_big : ε * (card α ^ 2 : ℕ) ≤ tris.card) :
+  G.far_from_triangle_free ε :=
+begin
+  rw far_from_triangle_free_iff,
+  introsI H _ hG hH,
+  rw ←nat.cast_sub (card_le_of_subset $ edge_finset_mono hG),
+  exact tris_big.trans
+    (nat.cast_le.2 $ far_from_triangle_free_of_disjoint_triangles_aux htris pd hG hH),
+end
+
+protected lemma edge_disjoint_triangles.far_from_triangle_free (hG : G.edge_disjoint_triangles)
+  (tris_big : ε * (card α ^ 2 : ℕ) ≤ (G.clique_finset 3).card) : G.far_from_triangle_free ε :=
+far_from_triangle_free_of_disjoint_triangles _ subset.rfl (by simpa using hG) tris_big
+
 variables [nonempty α]
 
+lemma far_from_triangle_free.lt_half (hG : G.far_from_triangle_free ε) : ε < 2⁻¹ :=
+begin
+  by_contra' hε,
+  have := hG.le_card_sub_card bot_le (clique_free_bot $ by norm_num),
+  simp only [set.to_finset_card (edge_set ⊥), fintype.card_of_finset, edge_set_bot, cast_zero,
+    finset.card_empty, tsub_zero] at this,
+  have hε₀ : 0 < ε := hε.trans_lt' (by norm_num),
+  rw inv_pos_le_iff_one_le_mul (zero_lt_two' 𝕜) at hε,
+  refine (this.trans $ le_mul_of_one_le_left (by positivity) hε).not_lt _,
+  rw [mul_assoc, mul_lt_mul_left hε₀],
+  norm_cast,
+  refine (mul_le_mul_left' (card_mono $ edge_finset_mono le_top) _).trans_lt _,
+  rw [edge_finset_top, filter_not, card_sdiff (subset_univ _), card_univ, sym2.card],
+  simp_rw [sym2.is_diag_iff_mem_range_diag, univ_filter_mem_range, mul_tsub,
+    nat.mul_div_cancel' (card α).even_mul_succ_self.two_dvd],
+  rw [card_image_of_injective _ sym2.diag_injective, card_univ, mul_add_one, two_mul, sq,
+    add_tsub_add_eq_tsub_right],
+  exact tsub_lt_self (mul_pos fintype.card_pos fintype.card_pos) fintype.card_pos,
+end
+
+lemma far_from_triangle_free.lt_one (hG : G.far_from_triangle_free ε) : ε < 1 :=
+hG.lt_half.trans $ by norm_num
+
 lemma far_from_triangle_free.nonpos (h₀ : G.far_from_triangle_free ε) (h₁ : G.clique_free 3) :
   ε ≤ 0 :=
 begin