A common variable minimax theorem for graphs

Abstract: Let

m a t h c a l G = G_{1} = (V, E_{1}), d o t s, 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}, d o t s, E_{m}

. Informally, a function

f : V i g h t a r r o w m a t h b b R

is smooth with respect to

G_{k} = (V, E_{k})

if

f (u) s i m f (v)

whenever

(u, v) i n E_{k}

. We study the problem of understanding whether there exists a nonconstant function that is smooth with respect to all graphs in

m a t h c a l G

, simultaneously, and how to find it if it exists.

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