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import FormalConjecturesUtilWritten on the Wall II - Conjecture 16
namespace WrittenOnTheWallII.GraphConjecture16
open Classical SimpleGraph
variable {α : Type*} [Fintype α] [DecidableEq α] [Nontrivial α]
WOWII Conjecture 16
For a simple connected graph G, the size b(G) of a largest induced bipartite subgraph
satisfies b(G) ≥ 2 * (rad(G) - 1) + max_{v ∈ V} l(v), where rad(G) is the radius
of G (the minimum eccentricity) and l(v) = indepNeighborsCard G v is the independence
number of the neighbourhood of v.
@[category research solved, AMS 5]
theorem conjecture16 (G : SimpleGraph α) (h : G.Connected) :
letI maxL := (Finset.univ.image (fun v => indepNeighborsCard G v)).max' (α:Type u_1inst✝²:Fintype αinst✝¹:DecidableEq αinst✝:Nontrivial αG:SimpleGraph αh:G.Connected⊢ (Finset.image (fun v => G.indepNeighborsCard v) Finset.univ).Nonempty All goals completed! 🐙)
letI r := G.radius.toNat
2 * ((r : ℝ) - 1) + (maxL : ℝ) ≤ b G := α:Type u_1inst✝²:Fintype αinst✝¹:DecidableEq αinst✝:Nontrivial αG:SimpleGraph αh:G.Connected⊢ 2 * (↑G.radius.toNat - 1) + ↑((Finset.image (fun v => G.indepNeighborsCard v) Finset.univ).max' ⋯) ≤ G.b
All goals completed! 🐙
-- Sanity checks
/-- The invariant `b G` (largest induced bipartite subgraph size) is nonneg. -/
@[category test, AMS 5]
example (G : SimpleGraph (Fin 3)) : 0 ≤ b G := Nat.cast_nonneg _
/-- In `K₂`, the max degree is 1. -/
@[category test, AMS 5]
example : (⊤ : SimpleGraph (Fin 2)).maxDegree = 1 := α:Type u_1inst✝²:Fintype αinst✝¹:DecidableEq αinst✝:Nontrivial α⊢ ⊤.maxDegree = 1 All goals completed! 🐙
end WrittenOnTheWallII.GraphConjecture16