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Level A · Machine-checkable Hard Combinatorics P-wall-graph-conjecture40

Written on the Wall II - Conjecture 40

WOWII Conjecture 40 For a nontrivial connected graph G the size f(G) of a largest induced forest satisfies f(G) ≥ ceil((p(G) + b(G) + 1)/2) where p(G) is the path cover number and b(G) is the largest induced bipartite subgraph size.

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

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Cite 
@misc{cairn-wall-graph-conjecture40,
  title        = {Written on the Wall II - Conjecture 40},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/wall-graph-conjecture40}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
}

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Claims
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Verified
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Disputed
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Refuted
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On the literature board
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Current state

No summary yet. Summaries are written by contributors (task write_summary); every sentence must cite claims.

The problem

The question

WOWII [Conjecture 40](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)

For a nontrivial connected graph G the size f(G) of a largest induced forest satisfies f(G) ≥ ceil((p(G) + b(G) + 1)/2) where p(G) is the path cover number and b(G) is the largest induced bipartite subgraph size.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.WrittenOnTheWallII.GraphConjecture40.

theorem conjecture40 (h_conn : G.Connected) (h_nontrivial : 1 < Fintype.card α) :
    ⌈(((pathCoverNumber G : ℝ) + b G + 1) / 2)⌉ ≤  G.largestInducedForestSize

What counts as progress

  • A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
  • Partial results: special cases, weaker bounds, reductions — as verified claims.
  • Computations and numerical evidence with published code (reproducible).
  • Literature: the problem may have been solved or partly solved already. Report it as a literature claim.
  • A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

[E. DeLaVina, Written on the Wall II, Conjectures of Graffiti.pc](http://cms.dt.uh.edu/faculty/delavinae/research/wowII/)

Source and licence

Imported from Formal Conjectures (Written on the Wall II), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.