GraphConjecture145.lean`

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8 declarations, 1 contain sorry, 0 declared as axioms

Written on the Wall II - Conjecture 145

The WOWII HTML uses $\lambda_{\min}(\overline{G})$ (the bar denotes graph complement). The formal statement below uses the local-independence minimum of $G^c$.

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

Definitions

The local independence minimum $\mathrm{lMin}(G)$ is: $$\mathrm{lMin}(G) = \min_{v \in V(G)} l(v)$$ where $l(v) = \mathrm{indepNeighborsCard}(G, v)$ is the independence number of the neighbourhood of $v$. This is the minimum over all vertices of the local independence number.

The boundary vertices $B(G)$ of a connected graph are the vertices $v$ such that the eccentricity of $v$ equals the diameter of $G$.

The eccentricity of a set $\mathrm{ecc}(S) = \max_{u \notin S} \min_{w \in S} \mathrm{dist}(u, w)$. In the conjecture below, $\mathrm{ecc}(B)$ is the eccentricity of the boundary set.

Conjecture 145: $\mathrm{tree}(G) \ge 2 \cdot \mathrm{ecc}(B) / \lambda_{\min}(\overline{G})$ where $\mathrm{tree}(G)$ is largestInducedTreeSize G, $\mathrm{ecc}(B)$ is the eccentricity of the boundary vertices, and $\lambda_{\min}(\overline{G})$ is the local independence minimum of the complement $\overline{G}$.

Imports (1)

FormalConjecturesUtil

def WrittenOnTheWallII.GraphConjecture145.localIndependenceMin localIndependenceMin G is the minimum over all vertices of the local independence number indepNeighborsCard G v line 61 source
Source documentation
localIndependenceMin G is the minimum over all vertices of the local independence number indepNeighborsCard G v. This equals $\mathrm{lMin}$ from DeLaVina's notation.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, Nontrivial, SimpleGraph, Nat
Proof uses: Finset.inf', Finset.univ, SimpleGraph.indepNeighborsCard
3 instances, coercions and auxiliary constants left out.
```
noncomputable def localIndependenceMin (G : SimpleGraph α) : ℕ :=
…
```
theorem WrittenOnTheWallII.GraphConjecture145.conjecture145 sorry line 88 source
depends on sorry what this means
Statement uses: Fintype, DecidableEq, Nontrivial, SimpleGraph, DecidableRel, SimpleGraph.Adj, SimpleGraph.Connected, LT.lt, Nat, OfNat.ofNat, WrittenOnTheWallII.GraphConjecture145.localIndependenceMin, Compl.compl, LE.le, HMul.hMul, SimpleGraph.eccSet, SimpleGraph.maxEccentricityVertices, SimpleGraph.largestInducedTreeSize
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
12 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
example WrittenOnTheWallII.GraphConjecture145.(example at line 98) line 98 source
Source documentation
largestInducedTreeSize is nonneg.
```
example (G : SimpleGraph (Fin 3)) : 0 ≤ largestInducedTreeSize G := Nat.zero_le _
…
```
example WrittenOnTheWallII.GraphConjecture145.(example at line 102) line 102 source
Source documentation
localIndependenceMin is nonneg (it is a natural number).
```
example (G : SimpleGraph (Fin 3)) : 0 ≤ localIndependenceMin G := Nat.zero_le _
…
```
Mentions: WrittenOnTheWallII.GraphConjecture145.localIndependenceMin
example WrittenOnTheWallII.GraphConjecture145.(example at line 106) line 106 source
Source documentation
For any graph on Fin 3, eccSet is nonneg.
```
example (G : SimpleGraph (Fin 3)) [DecidableRel G.Adj] : 0 ≤ eccSet G (maxEccentricityVertices G) :=
…
```
example WrittenOnTheWallII.GraphConjecture145.(example at line 111) line 111 source
Source documentation
maxEccentricityVertices is a subset of all vertices.
```
example (G : SimpleGraph (Fin 4)) : maxEccentricityVertices G ⊆ Set.univ := by
…
```
example WrittenOnTheWallII.GraphConjecture145.(example at line 117) line 117 source
Source documentation
localIndependenceMin G is at most indepNeighborsCard G v for any vertex v. This follows from the definition of inf' as the minimum.
```
example (G : SimpleGraph (Fin 4)) (v : Fin 4) :
    localIndependenceMin G ≤ indepNeighborsCard G v := by
…
```
Mentions: WrittenOnTheWallII.GraphConjecture145.localIndependenceMin
example WrittenOnTheWallII.GraphConjecture145.(example at line 125) line 125 source
Source documentation
localIndependenceMin G is a natural number, hence nonneg.
```
example (G : SimpleGraph (Fin 4)) : 0 ≤ localIndependenceMin G := Nat.zero_le _
…
```
Mentions: WrittenOnTheWallII.GraphConjecture145.localIndependenceMin