The Integer Domination Root Conjecture
Akbari, Alikhani, Oboudi and Peng conjectured in 2010 that 0 and -2 are the only integer roots of the domination polynomial , proven for trees and unicyclic graphs and verified exhaustively for small orders. The paper gives a counterexample of order 33 with an integer domination root at , built from an S-unit branch cancellation mechanism.
- Result
- Disproved
- Status
- Resolved
- AI contribution
- AI co-developed
- Method
- Construction
- Field
- Graph polynomials
- Posed by
- S. Akbari, S. Alikhani, M. R. Oboudi, Y.-H. Peng
- Year posed
- 2010
- Years open
- 16y
- Solved
- 2026-07-31
- Model
- Claude Fable 5
- Vendor
- Anthropic
- Collaborators
- Saeid Alikhani, Max Griswold
- Verification
- Unreviewed
- Publication
- Preprint
- Significance
- 11 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What the AI did
"The concept and theoretical formulation of the S-Unit Branch Cancellation mechanism were generated by Claude Fable 5." The authors verified all formal proofs and carried out independent computational validations. Alikhani is one of the conjecture's original posers.
Verification
Author-verified by exact enumeration, and one author co-posed the conjecture being refuted; no external review yet.