GraphConjecture198a.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 198a

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

Imports (1)

FormalConjecturesUtil

theorem WrittenOnTheWallII.GraphConjecture198a.conjecture198a WOWII Conjecture 198a For a simple connected graph G, if b(G) ≤ 2 + ecc_avg(G), then G has a Hamiltonian path sorry line 44 source
Source documentation
WOWII Conjecture 198a
For a simple connected graph G, if b(G) ≤ 2 + ecc_avg(G), then G has a Hamiltonian path. Here b(G) is the number of vertices in a largest induced bipartite subgraph, and ecc_avg(G) is the average eccentricity of G. A Hamiltonian path is a walk visiting every vertex exactly once.

L4DEL explanation
WOWII Conjecture 198a For a simple connected graph G, if b(G) ≤ 2 + ecc_avg(G), then G has a Hamiltonian path. Here b(G) is the number of vertices in a largest induced bipartite subgraph, and ecc_avg(G) is the average eccentricity of G. A Hamiltonian path is a walk visiting every vertex exactly once.
Sources and attribution: Pinned Lean source (1646ca16afd6)

depends on sorry what this means
Statement uses: Fintype, DecidableEq, Nontrivial, SimpleGraph, SimpleGraph.Connected, LE.le, Real, SimpleGraph.b, HAdd.hAdd, OfNat.ofNat, Nat, SimpleGraph.averageEccentricity, Exists, SimpleGraph.Walk, SimpleGraph.Walk.IsHamiltonian
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
16 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture198a (G : SimpleGraph α) (h : G.Connected)
    (hb : b G ≤ 2 + averageEccentricity G) :
    ∃ a b : α, ∃ p : G.Walk a b, p.IsHamiltonian := by
…
```
example WrittenOnTheWallII.GraphConjecture198a.(example at line 53) line 53 source
Source documentation
The averageEccentricity invariant is nonneg for any graph.
```
example (G : SimpleGraph (Fin 3)) : 0 ≤ averageEccentricity G := by
…
```
example WrittenOnTheWallII.GraphConjecture198a.(example at line 59) line 59 source
Source documentation
The invariant b G is nonneg.
```
example (G : SimpleGraph (Fin 3)) : 0 ≤ b G := Nat.cast_nonneg _
…
```