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Teschner conjectured that every finite simple graph $G$ with at least one edge satisfies $b(G) \leq \frac{3}{2}\Delta(G)$, where $b(G)$ is the bondage number and $\Delta(G)$ is the maximum degree. Yavari gives a connected cubic bipartite graph on 18 vertices with $b(G)=5$. Since $\Delta(G)=3$, this gives $b(G)=5 > \frac{3}{2}\Delta(G)=\frac{9}{2}$, providing a counterexample and disproving the conjecture.
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According to the author's disclosure and the publicly shared ChatGPT transcript, GPT-5.6 Sol Max was prompted to solve Teschner's conjecture and found an explicit counterexample. The resulting manuscript gives an 18-vertex connected cubic bipartite graph with bondage number 5, together with an exact finite certificate establishing the claimed bondage number. Yavari subsequently checked and wrote up the result.
Provider: OpenAI · Prompt public: unknown · Independence: unknown
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.
Correctness: unknown · statement fidelity: unaudited · peer review: none
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.
Source-reported tools: construction.
Independent: unknown · difference confidence: 0
VibeMath has not independently audited the mathematical statement, proof, or novelty claim.