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Problems / combinatorics

combinatorics / Graph theory

Written on the Wall II Conjecture 200

For every finite connected simple graph G with more than one vertex, does equality between the largest induced-tree order and the ceiling of one plus the average independence number of the open neighborhoods imply that G has a Hamiltonian path?

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55Significance / 100
1Frontier events
2Verification tasks
1Recorded attempts

Temporal state

Current frontier

Independently reproduced

Explicit 14-vertex counterexample independently reproduced under the frozen statement

Acceptance criteria: Confirm the frozen statement against a fresh primary-source or archival capture, replay the explicit graph independently, audit prior art, and obtain specialist review before asserting accepted resolution or priority.

Append-only history

Frontier timeline

combinatoricsAug 30, 2026Significance 68/100

WOWII Conjecture 200 counterexample independently reproduced

Prior state unknown→Explicit 14-vertex counterexample independently reproduced under the frozen statement

A clean standard-library implementation reconstructed the public 14-vertex graph and exhaustively checked all 16,383 nonempty induced vertex sets. It found largest induced-tree order 7, exact local-independence average 36/7, ceiling 7, and no Hamiltonian path. Two pinned candidate-supplied verifiers agreed on every published invariant.

Independently reproducedWritten on the Wall II Conjecture 200 →
SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review

Research memory

Claims and attempts

Scoped claims

Independently reproduced

The graph obtained from K_(6,8) by deleting two disjoint alternating six-cycles is a 14-vertex counterexample to the frozen Written on the Wall II Conjecture 200 implication: its largest induced-tree order and the ceiling of one plus its average open-neighborhood independence are both 7, while it has no Hamiltonian path.

The frozen statement and definitions are those recorded at the pinned source repository and Formal Conjectures snapshot. Open neighborhoods, induced-tree vertex count, exact rational averaging, and unrestricted Hamiltonian-path endpoints are used.

Recorded attempts

Independently reconstruct and exhaustively audit the WOWII-200 graph

success · MFN independent verifier

All three implementations agreed on 14 vertices, 36 edges, local-independence average 36/7, ceiling 7, largest induced-tree order 7, 78 maximum witnesses, and absence of a Hamiltonian path.

Evidence graph

Connected research record

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    The graph obtained from K_(6,8) by deleting two disjoint alternating six-cycles is a 14-vertex counterexample to the frozen Written on the Wall II Conjecture 200 implication: its largest induced-tree order and the ceilin

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