# Planar Graphs of Bounded Degree have Constant Queue Number

A queue layout of a graph consists of a linear order of its vertices and a partition of its edges into queues, so that no two independent edges of the same queue are nested. The queue number of a graph is the minimum number of queues required by any of its queue layouts. A long-standing conjecture by Heath, Leighton and Rosenberg states that the queue number of planar graphs is constant. This conjecture has been partially settled in the positive for several subfamilies of planar graphs (most of which have bounded treewidth). In this paper, we make a further important step towards settling this conjecture. We prove that planar graphs of bounded degree (which may have unbounded treewidth) have constant queue number. A notable implication of this result is that every planar graph of bounded degree admits a three-dimensional straight-line grid drawing in linear volume. Further implications are that every planar graph of bounded degree has constant track number, and that every k-planar graph (i.e., every graph that can be drawn in the plane with at most k crossings per edge) of bounded degree has constant queue number.

• 35 publications
• 15 publications
• 16 publications
• 20 publications
• 37 publications
• 7 publications
• 34 publications
08/31/2018

### Queue Layouts of Planar 3-Trees

A queue layout of a graph G consists of a linear order of the vertices o...
01/17/2019

### Queue Layouts of Graphs with Bounded Degree and Bounded Genus

We prove that graphs with bounded degree and bounded Euler genus have bo...
06/15/2021

### On the Queue Number of Planar Graphs

A k-queue layout is a special type of a linear layout, in which the line...
08/07/2017

### Layouts for Plane Graphs on Constant Number of Tracks

A k-track layout of a graph consists of a vertex k colouring, and a tota...
08/12/2020

### The Local Queue Number of Graphs with Bounded Treewidth

A queue layout of a graph G consists of a vertex ordering of G and a par...
09/04/2019

### On a Conjecture of Lovász on Circle-Representations of Simple 4-Regular Planar Graphs

Lovász conjectured that every connected 4-regular planar graph G admits ...
08/14/2019

### Light edges in 1-planar graphs of minimum degree 3

A graph is 1-planar if it can be drawn in the plane so that each edge is...