A Common Variable Minimax Theorem for Graphs

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