发表机构
Kitasato University（北里大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文否定Suzuki问题，证明除球面、射影平面和克莱因瓶外，每个闭曲面上存在具有两个弱不等价最优$1$-嵌入的图，并构造了指数多个弱不等价嵌入的图。
AI 中文摘要
我们研究了闭曲面上最优$1$-嵌入在弱等价意义下的唯一性问题。我们否定了Suzuki的一个问题，证明了对于除球面、射影平面和克莱因瓶之外的每个闭曲面，存在一个最优$1$-嵌入图，该图具有两个弱不等价的最优$1$-嵌入。我们的构造在环面上明确给出，然后通过适当的连通和构造扩展到其他曲面，该构造保持弱不等价性。作为推论，我们还得到随着亏格增加，具有指数多个两两弱不等价的最优$1$-嵌入的最优$1$-嵌入图。
英文摘要
We study the uniqueness of optimal $1$-embeddings on closed surfaces up to weak equivalence. Answering a question of Suzuki in the negative, we show that for every closed surface other than the sphere, the projective plane and the Klein bottle, there exists an optimal $1$-embedded graph admitting two weakly inequivalent optimal $1$-embeddings. Our construction is given explicitly on the torus and is then extended to other surfaces through a suitable connected-sum construction that preserves weak inequivalence. As a consequence, we also obtain optimal $1$-embedded graphs with exponentially many pairwise weakly inequivalent optimal $1$-embeddings as the genus increases.
