曲面共振图的Tratnik-Ye中位数猜想的一个证明
A Proof of the Tratnik-Ye Medianity Conjecture of Resonance Graphs on Surfaces
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中文总结 AI 辅助
本文证明了曲面共振图的Tratnik-Ye中位数猜想，并建立了到树的笛卡尔积的等距嵌入，给出了超立方体面奇偶嵌入的等距性判据。
中文摘要 AI 辅助
我们证明了Tratnik-Ye猜想：在闭曲面上，当允许的偶面构成所有面的真子集时，完美匹配的共振图的每个连通分量都是中位数图。嵌入不必是胞腔的或强的，图也不必是二部图。我们还证明了与每个面相关的商图是一棵树，并且这些商图给出了每个分量到树的笛卡尔积的等距嵌入。这为超立方体的面奇偶嵌入成为等距嵌入提供了一个判据。
英文摘要
We prove the Tratnik--Ye conjecture that every connected component of a resonance graph of perfect matchings on a closed surface is median whenever the allowed even faces form a proper subset of all faces. The embedding need not be cellular or strong, and the graph need not be bipartite. We also show that the quotient associated with each face is a tree and that these quotients give an isometric embedding of each component into a Cartesian product of trees. This yields a criterion for the facial parity embedding into a hypercube to be isometric.
发表机构
- School of Mathematics and Information Sciences, Yantai University（烟台大学数学与信息科学学院）
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