# Planar graphs have bounded queue-number

## TL;DR

This paper proves that all planar graphs have bounded queue-number by introducing layered partitions, and extends the result to all proper minor-closed classes, connecting structural graph properties with queue layouts.

## Contribution

It introduces layered partitions as a new structural tool and proves bounded queue-number for all proper minor-closed classes of graphs, resolving a longstanding conjecture.

## Key findings

- Planar graphs have bounded queue-number.
- Layered partitions relate to strong products and bounded treewidth.
- Proper minor-closed classes have bounded queue-number.

## Abstract

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.