Let \({\mathcal {G}} = \{G_1 = (V, E_1), \ldots , G_m = (V, E_m)\}\) be a collection of m graphs defined on a common set of vertices V but with different edge sets \(E_1, \ldots , E_m\). Informally, a function \(f :V \rightarrow {\mathbb {R}}\) is smooth with respect to \(G_k = (V,E_k)\) if \(f(u) \sim f(v)\) whenever \((u, v) \in E_k\). We study the problem of understanding whether there exists a nonconstant function that is smooth with respect to all graphs in \({\mathcal {G}}\), simultaneously, and how to find it if it exists.

中文翻译：

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图的通用变量极小极大定理

令\({\mathcal {G}} = \{G_1 = (V, E_1), \ldots , G_m = (V, E_m)\}\)是定义在一组公共顶点V上的m个图的集合，但是具有不同的边集\(E_1, \ldots , E_m\)。通俗地说，一个函数\(f :V \rightarrow {\mathbb {R}}\)对于\(G_k = (V,E_k)\ ) 是光滑的，如果\(f(u) \sim f(v)\ )每当\​​((u, v) \in E_k\)。我们研究了理解是否存在一个关于\({\mathcal {G}}\)中的所有图平滑的非常量函数，以及如果存在如何找到它的问题。