GraphConjecture100.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 100

Verbatim statement (WOWII #100, status O):

If G is a simple connected graph, then α(G) ≤ CEIL[(maximum of λ(v) + 0.5*length(Ḡ))/2]

Source: http://cms.uhd.edu/faculty/delavinae/research/wowII/all.html#conj100

The WOWII HTML uses length(Ḡ) (the bar denotes graph complement); the extracted JSON in our private repo previously dropped the overline. The formal statement below uses the Euclidean norm of the degree sequence of Gᶜ.

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

Definition of graph length

The WOWII definitions popup defines length(H) as the square root of the sum of the squares of the vertex degrees. This is degreeL2Norm H in Lean. Combined with the overline above, the inequality reads: α(G) ≤ ⌈(max_v l(v) + 0.5 · degreeL2Norm(Gᶜ)) / 2⌉ where l(v) = indepNeighbors G v.

Imports (1)

FormalConjecturesUtil

theorem WrittenOnTheWallII.GraphConjecture100.conjecture100 Documented declaration: conjecture100 sorry line 64 source
Source documentation
WOWII Conjecture 100 (status O):
For a simple connected graph G, α(G) ≤ ⌈(max_v l(v) + 0.5 · degreeL2Norm(Gᶜ)) / 2⌉ where α(G) = G.indepNum is the independence number, max_v l(v) is the maximum over all vertices of the independence number of the neighbourhood (in G), and degreeL2Norm(Gᶜ) is the square root of the sum of the squares of the degrees in the complement Gᶜ.

L4DEL explanation
WOWII Conjecture 100 (status O): For a simple connected graph G, α(G) ≤ ⌈(max_v l(v) + 0.5 · degreeL2Norm(Gᶜ)) / 2⌉ where α(G) = G.indepNum is the independence number, max_v l(v) is the maximum over all vertices of the independence number of the neighbourhood (in G), and degreeL2Norm(Gᶜ) is the square root of the sum of the squares of the degrees in the complement Gᶜ.
Sources and attribution: Pinned Lean source (1646ca16afd6)

depends on sorry what this means
Statement uses: Fintype, DecidableEq, Nontrivial, SimpleGraph, DecidableRel, SimpleGraph.Adj, SimpleGraph.Connected, Nat, Finset.max', Finset.image, SimpleGraph.indepNeighborsCard, Finset.univ, LE.le, Real, SimpleGraph.indepNum, Int.ceil, HDiv.hDiv, HAdd.hAdd, HMul.hMul, OfNat.ofNat and 2 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
52 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture100 (G : SimpleGraph α) [DecidableRel G.Adj] (h : G.Connected) :
    let maxL := (Finset.univ.image (indepNeighborsCard G)).max' (by simp)
…
```
example WrittenOnTheWallII.GraphConjecture100.(example at line 73) line 73 source
Source documentation
The independence number is nonneg.
```
example (G : SimpleGraph (Fin 3)) : 0 ≤ G.indepNum := Nat.zero_le _
…
```
example WrittenOnTheWallII.GraphConjecture100.(example at line 77) line 77 source
Source documentation
The Euclidean norm of the degree sequence is nonnegative.
```
example (G : SimpleGraph (Fin 2)) [DecidableRel G.Adj] : 0 ≤ degreeL2Norm G :=
…
```