GraphConjecture18.lean`

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

5 declarations, 1 contain sorry, 0 declared as axioms

Written on the Wall II - Conjecture 18

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

Imports (1)

FormalConjecturesUtil

theorem WrittenOnTheWallII.GraphConjecture18.conjecture18 WOWII Conjecture 18 For a simple connected graph sorry line 48 source
Source documentation
WOWII Conjecture 18
For a simple connected graph $G$, the size $b(G)$ of a largest induced bipartite subgraph satisfies $b(G) \ge \alpha(G) + \lceil \sqrt{\mathrm{dist}_{\max}(M)} \rceil$, where $\alpha(G)$ is the independence number, $M$ is the set of maximum-degree vertices, and $\mathrm{dist}_{\max}(M) = \max\{\mathrm{dist}_G(u,v) \mid u, v \in M\}$ is the maximum distance between two maximum-degree vertices (DeLaVina's dist_max(M)). Proven by Benny John (Feb. 2006), generalizing Schindl's proof of conjecture 17.

L4DEL explanation
WOWII Conjecture 18 For a simple connected graph $G$, the size $b(G)$ of a largest induced bipartite subgraph satisfies $b(G) \ge \alpha(G) + \lceil \sqrt{\mathrm{dist}_{\max}(M)} \rceil$, where $\alpha(G)$ is the independence number, $M$ is the set of maximum-degree vertices, and $\mathrm{dist}_{\max}(M) = \max\{\mathrm{dist}_G(u,v) \mid u, v \in M\}$ is the maximum distance between two maximum-degree vertices (DeLaVina's dist_max(M)). Proven by Benny John (Feb. 2006), generalizing Schindl's proof of conjecture 17.
Sources and attribution: Pinned Lean source (1646ca16afd6)

depends on sorry what this means
Statement uses: Fintype, DecidableEq, Nontrivial, SimpleGraph, DecidableRel, SimpleGraph.Adj, SimpleGraph.Connected, Set, Set.ofPred, Eq, Nat, SimpleGraph.degree, Membership.mem, SimpleGraph.neighborSet, SimpleGraph.maxDegree, LE.le, Real, HAdd.hAdd, SimpleGraph.indepNum, Int.ceil and 3 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
27 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture18 (G : SimpleGraph α) [DecidableRel G.Adj] (h : G.Connected) :
    let M : Set α := {v | G.degree v = G.maxDegree}
…
```
example WrittenOnTheWallII.GraphConjecture18.(example at line 57) line 57 source
Source documentation
The invariant b G is nonneg (cast of a natural number).
```
example (G : SimpleGraph (Fin 3)) : 0 ≤ b G := Nat.cast_nonneg _
…
```
example WrittenOnTheWallII.GraphConjecture18.(example at line 61) line 61 source
Source documentation
In $K_3$, the max degree is $2$.
```
example : (⊤ : SimpleGraph (Fin 3)).maxDegree = 2 := by decide
…
```
example WrittenOnTheWallII.GraphConjecture18.(example at line 65) line 65 source
Source documentation
eccSet G S is always nonneg.
```
example (G : SimpleGraph (Fin 3)) (S : Set (Fin 3)) : 0 ≤ eccSet G S := Nat.zero_le _
…
```
example WrittenOnTheWallII.GraphConjecture18.(example at line 69) line 69 source
Source documentation
distToSet G v S is always nonneg (it is a natural number).
```
example (G : SimpleGraph (Fin 3)) (v : Fin 3) (S : Set (Fin 3)) : 0 ≤ distToSet G v S :=
…
```