A common variable minimax theorem for graphs

Let $mathcal{G} = {G_1 = (V, E_1), dots, G_m = (V, E_m)}$ be a collection of $m$ graphs defined on a common set of vertices $V$ but with different edge sets $E_1, dots, E_m$. 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.