For $p\in \mathbb{N}$, 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. Centered colorings play an important role in the theory of sparse graph classes introduced by Nešetřil and Ossona de Mendez [31], [32], as they structurally characterize classes of bounded expansion — one of the key sparsity notions in this theory. More precisely, a class of graphs $\mathcal{C}$ has bounded expansion if and only if there is a function $f:\mathbb{N}\to \mathbb{N}$ such that every graph $G\in \mathcal{C}$ for every $p\in \mathbb{N}$ admits a p-centered coloring with at most $f(p)$ colors. Unfortunately, known proofs for the existence of such colorings yield large upper bounds on the function f governing the number of colors needed, even for as simple classes as planar graphs. 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 Σ 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 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 [1]. 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 $n^{\mathcal{O}(1)}$.

为了 $磷\in \mathbb{N}$，着色λ的曲线图的顶点的ģ是对中心，如果为每一个连通子ħ的ģ，无论是ħ接收到多于p下的色彩，λ或存在，在恰好出现一次的彩色ħ。中心着色在 Nešetřil 和 Ossona de Mendez [31]、[32] 引入的稀疏图类理论中发挥着重要作用，因为它们在结构上表征了有界扩展的类- 该理论中的关键稀疏性概念之一。更准确地说，一类图$\mathcal{C}$ 有界展开当且仅当存在函数 $F：\mathbb{N}\to \mathbb{N}$ 使得每个图 $G\in \mathcal{C}$ 对于每个 $磷\in \mathbb{N}$承认p中心着色最多$F(磷)$颜色。不幸的是，这种着色存在的已知证据在控制所需颜色数量的函数f上产生了很大的上限，即使对于像平面图这样简单的类也是如此。在本文中，我们证明了每${钾}_{吨}$-minor-free 图承认p中心着色$\mathcal{哦}({磷}^{G(吨)})$某些函数g 的颜色。在图可嵌入固定表面 Σ 的特殊情况下，我们证明它允许p中心着色$\mathcal{哦}({磷}^{19})$颜色，多项式的次数与 Σ 的属无关。这提供了从适当的次闭类绘制的图的p中心着色所需颜色数量的第一个多项式上限，这回答了 Dvořák [1] 提出的一个开放问题。作为一个算法应用，我们使用我们的主要结果来证明如果$\mathcal{C}$是固定的适当次闭类图，然后分别在p和n顶点上给定图H和G，其中$G\in \mathcal{C}$中，可以决定是否ħ是的一个子图ģ在时间$2^{\mathcal{哦}(磷\mathrm{日志}磷)}\cdot n^{\mathcal{哦}(1)}$ 和空间 $n^{\mathcal{哦}(1)}$.