Polynomial bounds for centered colorings on proper minor-closed graph classes
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For p∈N, a coloring λ of the vertices of a graph G is p-centered if for every connected subgraph H of G, either H receives more than p colors under λ or there is a color that appears exactly once in H. In this paper, we prove that every K_t-minor-free graph admits a p-centered coloring with O(p^g(t)) colors for some function g. In the special case that the graph is embeddable in a fixed surface Σ we show that it admits a p-centered coloring with O(p^19) colors, with the degree of the polynomial independent of the genus of Σ. This provides the first polynomial upper bounds on the number of colors needed in p-centered colorings of graphs drawn from proper minor-closed classes, which answers an open problem posed by Dvořák. As an algorithmic application, we use our main result to prove that if C is a fixed proper minor-closed class of graphs, then given graphs H and G, on p and n vertices, respectively, where G∈C, it can be decided whether H is a subgraph of G in time 2^O(p p)· n^O(1) and space n^O(1).
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