图中曲面嵌入的Fano框架
A Fano framework for embeddings of graphs in surfaces
AI总结：
本文提出Fano框架，将紧曲面上图胞腔嵌入的七个基本性质与Fano平面的点对应，利用射影子空间刻画性质组合，并借助中位图和扩展gem表示给出新刻画与统一结果。
摘要：
我们考虑紧曲面上图的胞腔嵌入的七个基本性质，并证明每个性质都可以与Fano平面$F$的一个点相关联，使得性质的允许组合对应于$F$的射影子空间。这个Fano框架使我们能够推导出涉及这七个性质的一系列蕴含关系，既提供了新结果，也统一了已有结果。对于每个性质，我们利用图嵌入的中位图，给出具有该性质的嵌入与$4$-正则图的一个相关结构之间的对应关系。我们应用这一对应关系来刻画一个图嵌入何时具有带其中一个性质的扭曲对偶。对于每个允许的性质组合，我们证明存在具有这些性质的图嵌入。我们研究了这七个性质与三个较弱的“欧拉”性质之间的联系。我们的证明涉及图嵌入的“gem”（图编码映射）表示的一个扩展版本中闭途径上的奇偶性条件。
英文摘要：
We consider seven fundamental properties of cellular embeddings of graphs in compact surfaces, and show that each property can be associated with a point of the Fano plane $F$, in such a way that allowable combinations of properties correspond to projective subspaces of $F$. This Fano framework allows us to deduce a number of implications involving the seven properties, providing new results and unifying existing ones. For each property, we provide a correspondence between embeddings with that property and an associated structure for $4$-regular graphs, using the medial graph of the graph embedding. We apply this to characterize when a graph embedding has a twisted dual with one of the properties. For each allowable combination of properties, we show that a graph embedding with these properties exists. We investigate connections between the seven properties and three weaker `Eulerian' properties. Our proofs involve parity conditions on closed walks in an extended version of the `gem' (graph-encoded map) representation of a graph embedding.
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