The defective chromatic number of a graph class is the infimum k such that there exists an integer d such that every graph in this class can be partitioned into at most k induced subgraphs with maximum degree at most d. Finding the defective chromatic number is a fundamental graph partitioning problem and received attention recently partially due to Hadwiger’s conjecture about coloring minor-closed families. In this paper, we prove that the defective chromatic number of any minor-closed family equals the simple lower bound obtained by the standard construction, confirming a conjecture of Ossona de Mendez, Oum, and Wood. This result provides the optimal list of unavoidable finite minors for infinite graphs that cannot be partitioned into a fixed finite number of induced subgraphs with uniformly bounded maximum degree. As corollaries about clustered coloring, we obtain a linear relation between the clustered chromatic number of any minor-closed family and the tree-depth of its forbidden minors, improving an earlier exponential bound proved by Norin, Scott, Seymour, and Wood and confirming the planar case of their conjecture.

图类的缺陷色数是下确界k，使得存在整数d，使得该类中的每个图可以划分为最多k 个诱导子图，最大度数至多为d。寻找有缺陷的色数是一个基本的图划分问题，最近受到关注，部分原因是哈维格关于对小闭族进行着色的猜想。在本文中，我们证明了任何小闭族的缺陷色数等于标准构造得到的简单下界，证实了 Ossona de Mendez、Oum 和 Wood 的猜想。该结果为无限图提供了不可避免的有限子图的最佳列表，该无限图无法划分为具有统一有界最大度的固定有限数量的诱导子图。作为关于聚类着色的推论，我们获得了任何次要封闭族的聚类色数与其禁止次要的树深度之间的线性关系，改进了 Norin、Scott、Seymour 和 Wood 证明的早期指数界限，并证实了他们猜想的平面情况。