AI 中文总结
本文证明了不存在非平凡的唯C_9-饱和图，通过按最长偶圈长度分类并辅以计算机枚举验证，解决了Wenger-West猜想在t=9的情形。
AI 中文摘要
图G是唯C_t-饱和的，如果G不包含长度为t的圈，并且对于补图中的每条边e，G+e恰好包含一个长度为t的圈；如果G至少有t个顶点，则称其为非平凡的。Wenger和West证明了不存在非平凡的唯C_6-或C_7-饱和图，并推测对于所有t≥6该结论同样成立；t=8的情形已被验证但从未发表，而t≥9的情形一直悬而未决。我们证明了t=9时的猜想：不存在非平凡的唯C_9-饱和图。证明过程是对最长偶圈长度L（L为4、6、8、10或12，且L≤12）进行情形分析。L=12、L=10和L=4的情形通过手工处理解决；对于L=6和L=8，通过手工对圈外分量进行分类，将每种情形归结为有界有限族配置，再通过简短、可重放的计算机枚举（带有已知答案的控制，并通过零命中计数保证完备性）予以消除。在此过程中，我们还以更精确的形式证明了Wenger和West未给出证明而陈述的结构引理在t=9时的实例。
英文摘要
A graph G is uniquely C_t-saturated if G contains no cycle of length t and, for every edge e of the complement, G+e contains exactly one cycle of length t; it is nontrivial if it has at least t vertices. Wenger and West proved that no nontrivial uniquely C_6- or C_7-saturated graphs exist, and conjectured the same for every t >= 6; the case t=8 was verified but never published, and t >= 9 has remained open. We prove the conjecture for t=9: there is no nontrivial uniquely C_9-saturated graph. The proof is a case analysis on the length L of a longest even cycle of length at most 12 (L is 4, 6, 8, 10, or 12). The cases L=12, L=10, and L=4 are settled by hand; for L=6 and L=8, hand classifications of the components outside the cycle reduce each case to a bounded finite family of configurations, eliminated by a short, replayable computer enumeration with known-answer controls and completeness certified by zero cap hits. Along the way we prove, in sharpened form, the t=9 instance of a structural lemma Wenger and West stated without proof.
Comments22 pages, 1 figure
