A common variable minimax theorem for graphs

Let G = G1 = (V, E1), dots, Gm = (V, Em) be a collection of m graphs defined on a common set of vertices V but with different edge sets E1, dots, Em. Informally, a function f :V rightarrow ℝ is smooth with respect to Gk = (V,Ek) if f(u) ~ f(v) whenever (u, v) ∈ Ek. We study the problem of understanding whether there exists a nonconstant function that is smooth with respect to all graphs in G, simultaneously, and how to find it if it exists.