Centered colorings in minor-closed graph classes

A vertex coloring \(\varphi\) of a graph \(G\) is \(p\)-centered if for every connected subgraph \(H\) of \(G\), either \(\varphi\) uses more than \(p\) colors on \(H\), or there is a color that appears exactly once on \(H\). We prove that for every fixed positive integer \(t\), every \(K_t\)-minor-free graph admits a \(p\)-centered coloring using \(\mathcal{O}(p^{t-1})\) colors.