AI 中文总结
本研究构造18顶点连通立方二分图作为反例，否定特施纳关于图的束缚数上界为3/2倍最大度的猜想，明确给出该反例的控制数、束缚数及验证细节。
AI 中文摘要
对于至少含一条边的有限简单图G，其束缚数b(G)是删除后能使控制数γ(G)增大的最少边数。特施纳猜想对所有图G，有b(G)≤(3/2)Δ(G)。我们通过构造一个18个顶点的连通立方二分图来否定该猜想，其γ(G)=6，b(G)=5。控制数通过二分划上的完全计数论证确定；显式的5条边删除操作可将控制数从6提升至7。为证明下界，我们给出一个精确有限证书：该图有297个最小控制集，删除其C(27,4)=17550个四元边子集的任意一个，都会至少留下一个控制集保持控制功能。该枚举是确定性的，仅使用精确整数和集合运算，且可通过附录中包含的完整标准库验证器复现。
英文摘要
For a finite simple graph $G$ with at least one edge, the bondage number $b(G)$ is the least number of edges whose deletion increases the domination number $γ(G)$. Teschner conjectured that $b(G)\le \tfrac32Δ(G)$ for every graph $G$. We disprove this conjecture by giving a connected cubic bipartite graph on eighteen vertices with \[ γ(G)=6 \qquad\text{and}\qquad b(G)=5. \] The domination number is established by a complete counting argument across the bipartition. An explicit five-edge deletion raises the domination number from six to seven. For the matching lower bound, we give an exact finite certificate: the graph has 297 minimum dominating sets, and deleting any one of its $\binom{27}{4}=17{,}550$ four-edge subsets leaves at least one of those sets dominating. The enumeration is deterministic, uses only exact integer and set operations, and is reproduced by the complete standard-library verifier included in the appendix.
Comments9 pages, no figures; complete Python 3 verifier included in the appendix
