On the Queue Number of Planar Graphs

TL;DR

This paper improves the upper bound on the queue number of planar graphs from 49 to 42 by refining existing techniques involving graph drawings, queue layouts, and graph decompositions.

Contribution

It presents the first significant improvement on the upper bound of the queue number of planar graphs since the previous result of 49.

Findings

Upper bound on queue number reduced from 49 to 42

Refined algorithms for graph drawings and layouts

Enhanced decomposition techniques for planar graphs

Abstract

A k-queue layout is a special type of a linear layout, in which the linear order avoids (k+1)-rainbows, i.e., k+1 independent edges that pairwise form a nested pair. The optimization goal is to determine the queue number of a graph, i.e., the minimum value of k for which a k-queue layout is feasible. Recently, Dujmovi\'c et al. [J. ACM, 67(4), 22:1-38, 2020] showed that the queue number of planar graphs is at most 49, thus settling in the positive a long-standing conjecture by Heath, Leighton and Rosenberg. To achieve this breakthrough result, their approach involves three different techniques: (i) an algorithm to obtain straight-line drawings of outerplanar graphs, in which the y-distance of any two adjacent vertices is 1 or 2, (ii) an algorithm to obtain 5-queue layouts of planar 3-trees, and (iii) a decomposition of a planar graph into so-called tripods. In this work, we push further each of these techniques to obtain the first non-trivial improvement on the upper bound from 49 to 42.