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.

翻译：kqueue 布局是一种特殊的线性布局类型, 线性顺序避免了( k+1 1) 彩虹布局, 即 k+1 独立的边缘, 形成一个嵌套配对。 优化的目标是确定一个图形的队列数, 即可以绘制 kqueue 布局的 k 最小值 。 最近, Dujmovi\' c et al. [ J. ACM, 67(4), 22:1- 38, 2020] 显示, 平面图的排队数最多为 49, 从而在正数中固定一个由 Heath、 L88on 和 Rosenberg 组成的长期直线性猜想。 为了实现这一突破性结果, 其方法涉及三种不同的技术 :(i) 获取外部平面图的直线图的算法, 其中任何两个相邻的脊的距离为 1 或 2, (ii) 获取平面 3 平面图的 5 队列布局布局的算法, 以及 (iii) 将平面图图图图的首部图解解成为42 。