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Teschner's Bondage-Number Conjecture

Teschner's universal bound b(G) <= (3/2)Delta(G) is false: the 18-vertex cubic bipartite graph has b(G) = 5 against a bound of 4.5. What survives is the restricted statement Teschner actually proved, that the bound holds for graphs of domination number at most three, and Gagarin and Zverovich's 2013 result that it holds for almost all graphs. The counterexample does not suggest a replacement bound, and the correct general upper bound for b(G) in terms of Delta(G) remains open.

Exact FrontierDelta

disproved→disproved

Scope and record

Occurred: Oct 29, 2026

Delta type: REGISTRY REVISION

Assumptions: VibeMathed verification: site-confirmed. Publication: preprint. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: Teschner's Bondage-Number Conjecture · Teschner bondage conjecture

Confidence: Not scored

Registry verification: site confirmed · preprint · resolved

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Attribution

VibeMathed
registry · event recorded by

GPT-5.6 Sol Max
model · ai model contributor · OpenAI

Yousof Yavari
human · human collaborator

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SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review

Named evidence

VibeMathed record: Teschner's Bondage-Number Conjecture

specialist registry · passed

Reproduced in full here on 13 August 2026. The counterexample is a single 18-vertex graph, so the claim is finite and was checked exhaustively rather than sampled. The edge list was transcribed from equation (3.1) and every quantity recomputed independently, without reading or running the author's verifier. Confirmed: connected, cubic and bipartite with the stated parts, 18 vertices and 27 distinct edges, so $\Delta(G) = 3$ ; domination number 6 by exhaustive search; and exactly 297 minimum dominating sets, the count the paper states, arrived at here independently. For the lower bound, all 20,853 edge subsets of size at most four were tested by the bundle criterion, and every one leaves at least one minimum dominating set intact, so $b(G) \ge 5$ . For the upper bound, deleting the five edges 0-6, 0-10, 0-16, 1-8 and 1-11 raises the domination number to 7, recomputed from scratch on the reduced graph rather than inferred from the criterion, so $b(G) \le 5$ . Therefore $b(G) = 5 > 4.5 = \frac32\Delta(G)$ and the conjecture is false. That enumeration is the entire mathematical content of the claim, so this is a complete independent check. Caveats: the preprint is two days old, is hosted on figshare rather than arXiv, and has no peer review; the acknowledgements name Eric Hou (UBC) as an independent verifier, but that is a private check, not a public endorsement by a specialist.