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module
public import FormalConjecturesUtilWritten on the Wall II - Conjecture 61
Reference: E. DeLaVina, Written on the Wall II, Conjectures of Graffiti.pc
@[expose] public sectionnamespace WrittenOnTheWallII.GraphConjecture61open SimpleGraphvariable {α : Type*} [Fintype α] [DecidableEq α] [Nontrivial α]WOWII Conjecture 61
For a simple connected graph $G$, the size $f(G)$ of a largest induced forest satisfies $f(G) \ge \mathrm{residue}(G) + \lceil \mathrm{diam}(G) / 3 \rceil$, where $\mathrm{residue}(G)$ is the Havel-Hakimi residue and $\mathrm{diam}(G)$ is the diameter of $G$.
See: Favaron, Mahéo, Saclé (1991) for the residue; DeLaVina's Graffiti.pc for the conjecture.
@[category research open, AMS 5]
theorem conjecture61 (G : SimpleGraph α) [DecidableRel G.Adj] (h : G.Connected) :
(residue G : ℝ) + ⌈(G.diam : ℝ) / 3⌉ ≤ (G.largestInducedForestSize : ℝ) := α:Type u_1inst✝³:Fintype αinst✝²:DecidableEq αinst✝¹:Nontrivial αG:SimpleGraph αinst✝:DecidableRel G.Adjh:G.Connected⊢ ↑G.residue + ↑⌈↑G.diam / 3⌉ ≤ ↑G.largestInducedForestSize
All 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 := α:Type u_1inst✝²:Fintype αinst✝¹:DecidableEq αinst✝:Nontrivial α⊢ ⊤.residue = 1
All goals completed! 🐙end WrittenOnTheWallII.GraphConjecture61