Planar Graphs have Bounded Queue-Number
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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 H-partitions, and the result that every planar graph has such a partition of bounded layered width in which H has bounded treewidth. These results generalise for graphs of bounded Euler genus. 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.
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