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.

中文翻译：

---

关于平面图的队列数

k-queue 布局是一种特殊类型的线性布局，其中线性顺序避免了 (k+1)-rainbows，即 k+1 条独立边成对形成嵌套对。优化目标是确定图的队列数，即k-queue布局可行的k的最小值。最近，Dujmovi\'c 等人。[J. ACM, 67(4), 22:1-38, 2020] 表明平面图的队列数最多为 49，从而解决了 Heath、Leighton 和 Rosenberg 长期存在的猜想。为了实现这一突破性结果，他们的方法涉及三种不同的技术：(i) 一种获得外平面图直线图的算法，其中任意两个相邻顶点的 y 距离为 1 或 2，(ii) 一种算法获得平面 3 棵树的 5 队列布局，(iii) 将平面图分解为所谓的三脚架。在这项工作中，我们进一步推动了这些技术中的每一项，以获得从 49 到 42 的上限的第一个非平凡改进。