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import FormalConjecturesUtilWritten on the Wall II - Conjecture 18
Reference: E. DeLaVina, Written on the Wall II, Conjectures of Graffiti.pc
namespace WrittenOnTheWallII.GraphConjecture18open SimpleGraphvariable {α : Type*} [Fintype α] [DecidableEq α] [Nontrivial α]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.
@[category research solved, AMS 5]
theorem conjecture18 (G : SimpleGraph α) [DecidableRel G.Adj] (h : G.Connected) :
let M : Set α := {v | G.degree v = G.maxDegree}
(G.indepNum : ℝ) + ⌈Real.sqrt (distMaxSet G M : ℝ)⌉ ≤ b G := α:Type u_1inst✝³:Fintype αinst✝²:DecidableEq αinst✝¹:Nontrivial αG:SimpleGraph αinst✝:DecidableRel G.Adjh:G.Connected⊢ let M := {v | G.degree v = G.maxDegree};
↑α(G) + ↑⌈√↑(G.distMaxSet M)⌉ ≤ G.b
All goals completed! 🐙-- Sanity checks
The invariant b G is nonneg (cast of a natural number).
@[category test, AMS 5]
example (G : SimpleGraph (Fin 3)) : 0 ≤ b G := Nat.cast_nonneg _In $K_3$, the max degree is $2$.
@[category test, AMS 5]
example : (⊤ : SimpleGraph (Fin 3)).maxDegree = 2 := α:Type u_1inst✝²:Fintype αinst✝¹:DecidableEq αinst✝:Nontrivial α⊢ ⊤.maxDegree = 2 All goals completed! 🐙
eccSet G S is always nonneg.
@[category test, AMS 5]
example (G : SimpleGraph (Fin 3)) (S : Set (Fin 3)) : 0 ≤ eccSet G S := Nat.zero_le _
distToSet G v S is always nonneg (it is a natural number).
@[category test, AMS 5]
example (G : SimpleGraph (Fin 3)) (v : Fin 3) (S : Set (Fin 3)) : 0 ≤ distToSet G v S :=
Nat.zero_le _end WrittenOnTheWallII.GraphConjecture18