Polynomial bounds for centered colorings on proper minor-closed graph classes

For p∈N, a coloring λ of the vertices graph G is p-centered if for every connected subgraph H G, either receives more than p colors under or there color that appears exactly once in H. Centered colorings play an important role theory sparse classes introduced by Nešetřil and Ossona de Mendez [31], [32], as they structurally characterize bounded expansion — one key sparsity notions this theory. More precisely, class graphs C has only function f:N→N such G∈C p∈N admits with at most f(p) colors. Unfortunately, known proofs existence yield large upper bounds on f governing number needed, even simple planar graphs. In paper, we prove Kt-minor-free O(pg(t)) some g. special case embeddable fixed surface Σ show it O(p19) colors, degree polynomial independent genus Σ. This provides first needed drawn from proper minor-closed classes, which answers open problem posed Dvořák [1]. As algorithmic application, use our main result to graphs, then given n vertices, respectively, where G∈C, can be decided whether time 2O(plog⁡p)⋅nO(1) space nO(1).