FormalConjectures.WrittenOnTheWallII.GraphConjecture13

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import FormalConjecturesUtil

Written on the Wall II - Conjecture 13

Reference: E. DeLaVina, Written on the Wall II, Conjectures of Graffiti.pc

namespace WrittenOnTheWallII.GraphConjecture13open SimpleGraphvariable {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' (byα: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 simpAll goals completed! 🐙)
    (G.diam : ℝ) + (maxL : ℝ) - 1 ≤ b G := byα: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
  sorryAll 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 := byα:Type u_1inst✝²:Fintype αinst✝¹:DecidableEq αinst✝:Nontrivial α⊢ ⊤.maxDegree = 1 decide +nativeAll goals completed! 🐙

end WrittenOnTheWallII.GraphConjecture13