We show that planar graphs have bounded queue-number, thus proving a conjecture of Heath, Leighton and Rosenberg from 1992. The key to the proof is a new structural tool called layered partitions, and the result that every planar graph has a vertex-partition and a layering, such that each part has a bounded number of vertices in each layer, and the quotient graph has bounded treewidth. This result generalises for graphs of bounded Euler genus. Moreover, we prove that every graph in a minor-closed class has such a layered partition if and only if the class excludes some apex graph. Building on this work and using the graph minor structure theorem, we prove that every proper minor-closed class of graphs has bounded queue-number. Layered partitions have strong connections to other topics, including the following two examples. First, they can be interpreted in terms of strong products. We show that every planar graph is a subgraph of the strong product of a path with some graph of bounded treewidth. Similar statements hold for all proper minor-closed classes. Second, we give a simple proof of the result by DeVos et al. (2004) that graphs in a proper minor-closed class have low treewidth colourings.

中文翻译：

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平面图有界队列数

我们证明平面图有界队列数，从而证明了 Heath、Leighton 和 Rosenberg 于 1992 年的猜想。证明的关键是一种称为分层分区的新结构工具，结果每个平面图都有一个顶点分区和一个分层，这样每个部分在每一层中都有一个有限数量的顶点，并且商图有一个有界树宽。这个结果推广到有界 Euler 属的图。此外，我们证明，当且仅当该类排除某个顶点图时，小闭类中的每个图都具有这样的分层分区。在这项工作的基础上并使用图次要结构定理，我们证明了每个适当的次要闭类图都具有有界队列数。分层分区与其他主题有很强的联系，包括以下两个示例。第一的，它们可以解释为强大的产品。我们证明每个平面图都是路径与某个有界树宽图的强乘积的子图。类似的陈述适用于所有适当的次要封闭类。其次，我们对 DeVos 等人的结果进行了简单的证明。(2004) 认为适当的小封闭类中的图具有低树宽着色。