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

For $p\in \mathbb{N}$, a coloring $\lambda$ of the vertices of a graph $G$ is {\em{$p$-centered}} if for every connected subgraph~$H$ of $G$, either $H$ receives more than $p$ colors under $\lambda$ 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 $\mathcal{O}(p^{g(t)})$ colors for some function $g$. In the special case that the graph is embeddable in a fixed surface $\Sigma$ we show that it admits a $p$-centered coloring with $\mathcal{O}(p^{19})$ colors, with the degree of the polynomial independent of the genus of $\Sigma$. 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 Dvo\v{r}{\'a}k. As an algorithmic application, we use our main result to prove that if $\mathcal{C}$ is a fixed proper minor-closed class of graphs, then given graphs $H$ and $G$, on $p$ and $n$ vertices, respectively, where $G\in \mathcal{C}$, it can be decided whether $H$ is a subgraph of $G$ in time $2^{{\mathcal{O}(p\log} p)}\cdot n^{{\mathcal{O}(1)}$} and space $n^{{\mathcal{O}(1)}$.}