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import FormalConjecturesUtilWritten on the Wall II - Conjecture 103
Reference: E. DeLaVina, Written on the Wall II, Conjectures of Graffiti.pc
namespace WrittenOnTheWallII.GraphConjecture103open SimpleGraphThe 11-vertex counterexample: a triangle with four leaves on each of two vertices.
abbrev wowii103Counterexample : SimpleGraph (Fin 11) :=
SimpleGraph.fromEdgeSet {
s(0, 1), s(0, 2), s(1, 2),
s(0, 3), s(0, 4), s(0, 5), s(0, 6),
s(1, 7), s(1, 8), s(1, 9), s(1, 10)
}The counterexample is connected.
@[category test, AMS 5]
theorem wowii103Counterexample_connected : wowii103Counterexample.Connected := ⊢ wowii103Counterexample.Connected
All goals completed! 🐙The counterexample has independence number nine.
⊢ wowii103Counterexample.computable_indep_num = 9
All goals completed! 🐙The largest induced bipartite subgraph of the counterexample has ten vertices.
⊢ wowii103Counterexample.computableLargestInducedBipartiteSubgraphSize = 10
All goals completed! 🐙The counterexample has average eccentricity $30/11$.
hsum:∑ v, (wowii103Counterexample.eccent v).toNat = 30⊢ ↑30 / ↑(Fintype.card (Fin 11)) = 30 / 11
All goals completed! 🐙The logarithm of the counterexample's average eccentricity lies strictly between one and two.
⊢ Real.exp 1 < 30 / 11
exact Real.exp_one_lt_d9.trans (⊢ 2.7182818286 < 30 / 11 All goals completed! 🐙)
⊢ Real.log (30 / 11) < 2 exact (Real.log_lt_sub_one_of_pos (⊢ 0 < 30 / 11 All goals completed! 🐙) (⊢ 30 / 11 ≠ 1 All goals completed! 🐙)).trans (⊢ 30 / 11 - 1 < 2 All goals completed! 🐙)WOWII Conjecture 103
For a simple connected graph $G$,
$\alpha(G) \le \lfloor b(G) - \ln(\mathrm{ecc_avg}(G)) \rfloor$
where $\alpha(G) = G.\mathrm{indepNum}$ is the independence number,
$b(G)$ is the largest induced bipartite subgraph size, and
$\mathrm{ecc_avg}(G) = G.\mathrm{averageEccentricity}$ is the average
eccentricity of $G$. Uses Real.log (natural logarithm).
This conjecture is false. The graph wowii103Counterexample has independence number $9$,
largest induced bipartite subgraph size $10$, and average eccentricity $30/11$. Since
$1 < \ln(30/11) < 2$, the proposed upper bound is $8$.
h:∀ (α : Type) [inst : Fintype α] [DecidableEq α] [Nontrivial α] (G : SimpleGraph α),
G.Connected → ↑α(G) ≤ ↑⌊G.b - Real.log G.averageEccentricity⌋hbad:↑9 ≤ ↑⌊↑10 - Real.log (30 / 11)⌋hfloor:⌊10 - Real.log (30 / 11)⌋ = 8⊢ False
All goals completed! 🐙end WrittenOnTheWallII.GraphConjecture103