FormalConjectures.WrittenOnTheWallII.GraphConjecture59

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import FormalConjecturesUtil

Written on the Wall II - Conjecture 59

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


namespace WrittenOnTheWallII.GraphConjecture59


open Classical SimpleGraph


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

WOWII Conjecture 59

For a simple connected graph $G$, the size $f(G)$ of a largest induced forest satisfies $f(G) \ge \lceil \sqrt{\mathrm{residue}(G) \cdot b(G)} \rceil$, where $\mathrm{residue}(G)$ is the Havel-Hakimi residue (the number of zeros remaining after applying the Havel-Hakimi algorithm to the degree sequence until termination) and $b(G)$ is the size of a largest induced bipartite subgraph.

See: Favaron, Mahéo, Saclé (1991) for the residue; DeLaVina's Graffiti.pc for the conjecture.

@[category research open, AMS 5]
theorem declaration uses 'sorry'conjecture59 (G : SimpleGraph α) [DecidableRel G.Adj] (h : G.Connected) :
    ⌈Real.sqrt ((residue G : ℝ) * b G)⌉ ≤ (G.largestInducedForestSize : ℝ) := byα:Type u_1inst✝³:Fintype αinst✝²:DecidableEq αinst✝¹:Nontrivial αG:SimpleGraph αinst✝:DecidableRel G.Adjh:G.Connected⊢ ↑⌈√(↑G.residue * G.b)⌉ ≤ ↑G.largestInducedForestSize
  sorryAll goals completed! 🐙


-- Sanity checks

/-- The `largestInducedForestSize` is nonneg. -/
@[category test, AMS 5]
example (G : SimpleGraph (Fin 3)) : 0 ≤ G.largestInducedForestSize := Nat.zero_le _


/-- 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 := byα:Type u_1inst✝²:Fintype αinst✝¹:DecidableEq αinst✝:Nontrivial α⊢ ⊤.residue = 1
  unfold residueα:Type u_1inst✝²:Fintype αinst✝¹:DecidableEq αinst✝:Nontrivial α⊢ residueAux ((Multiset.map (fun v => ⊤.degree v) Finset.univ.val).sort fun x1 x2 => x1 ≥ x2) = 1; decide +nativeAll goals completed! 🐙


end WrittenOnTheWallII.GraphConjecture59