GraphConjecture133.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 133

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

Imports (1)

FormalConjecturesUtil

theorem WrittenOnTheWallII.GraphConjecture133.conjecture133 WOWII Conjecture 133: For a simple connected graph sorry line 48 source
Source documentation
WOWII Conjecture 133:
For a simple connected graph $G$, $\operatorname{path}(G) \ge \operatorname{rad}(G) + (\mathrm{avg}_v\, l(v))^{cC_4(G)}$, where $\operatorname{path}(G)$ is the path number of the graph (number of vertices of a largest induced path), $\operatorname{rad}(G)$ is the radius (minimum eccentricity, as a natural number), $\mathrm{avg}_v\, l(v) = l(G)$ is the average independence number of vertex neighbourhoods, and $cC_4(G)$ is the $C_4$-free characteristic function (1 if $G$ is $C_4$-free, not necessarily induced, and 0 otherwise).

L4DEL explanation
WOWII Conjecture 133: For a simple connected graph $G$, $\operatorname{path}(G) \ge \operatorname{rad}(G) + (\mathrm{avg}_v\, l(v))^{cC_4(G)}$, where $\operatorname{path}(G)$ is the path number of the graph (number of vertices of a largest induced path), $\operatorname{rad}(G)$ is the radius (minimum eccentricity, as a natural number), $\mathrm{avg}_v\, l(v) = l(G)$ is the average independence number of vertex neighbourhoods, and $cC_4(G)$ is the $C_4$-free characteristic function (1 if $G$ is $C_4$-free, not necessarily induced, and 0 otherwise).
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, ENat.toNat, SimpleGraph.radius, Exists, And, Ne, ite, Eq, OfNat.ofNat, LE.le, Real, HAdd.hAdd, HPow.hPow and 2 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
26 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture133 (G : SimpleGraph α) [DecidableRel G.Adj] (h : G.Connected) :
    let rad := G.radius.toNat
…
```
example WrittenOnTheWallII.GraphConjecture133.(example at line 60) line 60 source
Source documentation
The path G invariant is nonneg.
```
example (G : SimpleGraph (Fin 3)) : 0 ≤ (path G : ℝ) := Nat.cast_nonneg _
…
```
example WrittenOnTheWallII.GraphConjecture133.(example at line 64) line 64 source
Source documentation
radius toNat is nonneg.
```
example (G : SimpleGraph (Fin 3)) : 0 ≤ G.radius.toNat := Nat.zero_le _
…
```