Planar Graphs Have Bounded Queue-Number

• Published in: Journal of the ACM Vol. 67; no. 4; pp. 1 - 38
• Main Authors: Dujmović, Vida, Joret, Gwenaël, Micek, Piotr, Morin, Pat, Ueckerdt, Torsten, Wood, David R.
• Format: Journal Article
• Language: English
• Published: New York Association for Computing Machinery 01.08.2020
• Subjects: Apexes; Graph theory; Graphs; Partitions; Queues; Quotients
• ISSN: 0004-5411; 1557-735X
• DOI: 10.1145/3385731

We show that planar graphs have bounded queue-number, thus proving a conjecture of Heath et al. [66] 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. [31] that graphs in a proper minor-closed class have low treewidth colourings.