Minor-Universal Graph for Graphs on Surfaces
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We show that, for every n and every surface Σ, there is a graph U embeddable on Σ with at most cn^2 vertices that contains as minor every graph embeddable on Σ with n vertices. The constant c depends polynomially on the Euler genus of Σ. This generalizes a well-known result for planar graphs due to Robertson, Seymour, and Thomas [Quickly Excluding a Planar Graph. J. Comb. Theory B, 1994] which states that the square grid on 4n^2 vertices contains as minor every planar graph with n vertices.
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