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import FormalConjecturesUtilWritten on the Wall II - Conjecture 13
namespace WrittenOnTheWallII.GraphConjecture13
open Classical SimpleGraph
variable {α : Type*} [Fintype α] [DecidableEq α] [Nontrivial α]
WOWII Conjecture 13
For a simple connected graph G, the size b(G) of a largest induced bipartite subgraph
satisfies b(G) ≥ diam(G) + max_{v ∈ V} l(v) - 1, where diam(G) is the diameter
of G and l(v) = indepNeighborsCard G v is the independence number of the
neighbourhood of v.
@[category research solved, AMS 5]
theorem conjecture13 (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! 🐙)
(G.diam : ℝ) + (maxL : ℝ) - 1 ≤ b G := α:Type u_1inst✝²:Fintype αinst✝¹:DecidableEq αinst✝:Nontrivial αG:SimpleGraph αh:G.Connected⊢ ↑G.diam + ↑((Finset.image (fun v => G.indepNeighborsCard v) Finset.univ).max' ⋯) - 1 ≤ G.b
All goals completed! 🐙
-- Sanity checks
/-- The largest induced bipartite subgraph invariant `b G` is nonneg for any graph. -/
@[category test, AMS 5]
example (G : SimpleGraph (Fin 3)) : 0 ≤ b G := Nat.cast_nonneg _
/-- In `K₂`, all vertices have degree 1, so `indepNeighborsCard` at vertex 0 is 0
(the neighbourhood of 0 consists of only vertex 1, which has no independent edges). -/
@[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.GraphConjecture13