FormalConjectures.WrittenOnTheWallII.GraphConjecture109

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import FormalConjecturesUtil

Written on the Wall II - Conjecture 109

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


namespace WrittenOnTheWallII.GraphConjecture109


open Classical SimpleGraph

WOWII Conjecture 109

For a simple connected graph $G$, the independence number $\alpha(G)$ was conjectured to satisfy $\alpha(G) \le \lfloor (\mathrm{residue}(G) + 2 \cdot b(G)) / 3 \rfloor$, where $\mathrm{residue}(G)$ is the Havel--Hakimi residue and $b(G)$ is the size of a largest induced bipartite subgraph.

This is false. A connected graph on 21 vertices has an independent set of size 15, residue 8, and no induced bipartite subgraph with more than 18 vertices, so the conjectured right-hand side is at most 14.

@[category research solved, AMS 5,
  formal_proof using formal_conjectures at "https://github.com/DomTheDeveloper/formal-conjectures/blob/cf59008ef1cd432bf9803275dcf5d62ab1f094a3/FormalConjectures/WrittenOnTheWallII/GraphConjecture109.lean"]
theorem declaration uses 'sorry'conjecture109 : answer(False) ↔
    ∀ (α : Type) [Fintype α] [DecidableEq α] [Nontrivial α]
      (G : SimpleGraph α) [DecidableRel G.Adj] (_h : G.Connected),
      (G.indepNum : ℝ) ≤ ⌊((residue G : ℝ) + 2 * b G) / 3⌋ := by⊢ False ↔
  ∀ (α : Type) [inst : Fintype α] [DecidableEq α] [Nontrivial α] (G : SimpleGraph α) [inst_3 : DecidableRel G.Adj],
    G.Connected → ↑α(G) ≤ ↑⌊(↑G.residue + 2 * G.b) / 3⌋
  sorryAll goals completed! 🐙


-- Sanity checks

/-- The invariant `b G` is nonneg (cast of a natural number). -/
@[category test, AMS 5]
example (G : SimpleGraph (Fin 3)) : 0 ≤ b G := Nat.cast_nonneg _


/-- The residue of $K_2$ equals $1$: degree sequence is $[1, 1]$; one Havel-Hakimi
step gives $[0]$, leaving a single zero. -/
@[category test, AMS 5]
example : residue (⊤ : SimpleGraph (Fin 2)) = 1 := by⊢ ⊤.residue = 1
  unfold residue⊢ residueAux ((Multiset.map (fun v => ⊤.degree v) Finset.univ.val).sort fun x1 x2 => x1 ≥ x2) = 1
  decide +nativeAll goals completed! 🐙


end WrittenOnTheWallII.GraphConjecture109