GraphConjecture61.lean`

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Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

3 declarations, 1 contain sorry, 0 declared as axioms

Written on the Wall II - Conjecture 61

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

Imports (1)

FormalConjecturesUtil

theorem WrittenOnTheWallII.GraphConjecture61.conjecture61 WOWII Conjecture 61 For a simple connected graph sorry line 46 source
Source documentation
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.

L4DEL explanation
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.
Sources and attribution: Pinned Lean source (1646ca16afd6)

depends on sorry what this means
Statement uses: Fintype, DecidableEq, Nontrivial, SimpleGraph, DecidableRel, SimpleGraph.Adj, SimpleGraph.Connected, LE.le, Real, HAdd.hAdd, SimpleGraph.residue, Int.ceil, HDiv.hDiv, SimpleGraph.diam, OfNat.ofNat, Nat, SimpleGraph.largestInducedForestSize
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
34 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture61 (G : SimpleGraph α) [DecidableRel G.Adj] (h : G.Connected) :
    (residue G : ℝ) + ⌈(G.diam : ℝ) / 3⌉ ≤ (G.largestInducedForestSize : ℝ) := by
…
```
example WrittenOnTheWallII.GraphConjecture61.(example at line 54) line 54 source
Source documentation
The largestInducedForestSize is nonneg.
```
example (G : SimpleGraph (Fin 3)) : 0 ≤ G.largestInducedForestSize := Nat.zero_le _
…
```
example WrittenOnTheWallII.GraphConjecture61.(example at line 59) line 59 source
Source documentation
The residue of $K_2$ equals $1$: degree sequence is $[1, 1]$; one Havel-Hakimi step gives $[0]$, leaving a single zero.
```
example : residue (⊤ : SimpleGraph (Fin 2)) = 1 := by
…
```