FormalConjectures.WrittenOnTheWallII.GraphConjecture58

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import FormalConjecturesUtil

Written on the Wall II - Conjecture 58

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

Counterexample

The conjecture is false. A counterexample is given by taking $K_{3,3}$ (with bipartition ${0,1,2}$, ${3,4,5}$) and $K_{73}$ (on vertices ${6,\ldots,78}$), then adding edges between vertex $0$ and every vertex of $K_{73}$.

This graph $G$ has $n = 79$ vertices and satisfies:

$b(G) \ge 7$: the set ${0,1,2,3,4,5,6}$ induces a bipartite subgraph

$f(G) \le 6$: the largest induced forest has at most 6 vertices

$l_{\mathrm{avg}}(G) = 92/79$

$\lceil b(G) / l_{\mathrm{avg}}(G) \rceil \ge 7 > 6 \ge f(G)$


namespace WrittenOnTheWallII.GraphConjecture58


open SimpleGraph Finset

WOWII Conjecture 58

For a connected graph G, the size f(G) of a largest induced forest satisfies f(G) ≥ ceil( b(G) / average l(v) ) where b(G) is the largest induced bipartite subgraph and l(v) is the independence number of G.neighborSet v.

This conjecture is false. A counterexample is the graph described in the module docstring above: a $K_{3,3}$ joined to a $K_{73}$ via vertex $0$, giving $\lceil b/l_{\mathrm{avg}} \rceil \ge 7 > 6 \ge f(G)$.

The counterexample has been found by Moritz Firsching and Goran Žužić using an experimental pipeline.

@[category research solved, AMS 5, formal_proof using formal_conjectures at
"https://github.com/mo271/formal-conjectures/blob/4bd72a06842a10e1b8d7bb0fd6b1ef5e6bd20210/FormalConjectures/WrittenOnTheWallII/GraphConjecture58.lean#L772"]
theorem declaration uses 'sorry'conjecture58 : answer(False) ↔
    ∀ (α : Type) [Fintype α] [DecidableEq α] [Nontrivial α]
      (G : SimpleGraph α) [DecidableRel G.Adj] (_hG : G.Connected),
      Nat.ceil (G.b / G.l_avg) ≤ G.largestInducedForestSize := by⊢ False ↔
  ∀ (α : Type) [inst : Fintype α] [DecidableEq α] [Nontrivial α] (G : SimpleGraph α) [DecidableRel G.Adj],
    G.Connected → ⌈G.b / G.l_avg⌉₊ ≤ G.largestInducedForestSize
  sorryAll goals completed! 🐙


end WrittenOnTheWallII.GraphConjecture58