AI 中文总结
本文指出Bekos等人提出的将平面图队列数上界降至42的算法存在漏洞，其所需的某一选择无法保证存在，故该上界未获证明。
AI 中文摘要
图的队列布局由顶点的线性顺序和边划分为若干队列构成，使得同一队列中的两条边互不嵌套。图的队列布局所需的最少队列数称为其队列数。平面乘积结构定理指出，每个平面图都是简单树宽至多为3的图、完全图K₃和路径的强乘积的子图。该强乘积的队列布局可使用49个队列（Wood，2005），这意味着平面图的队列数至多为49。近来，Bekos、Gronemann和Raftopoulou（《Algorithmica》，2023）研究了如何优化基于乘积结构的通用方法以应用于平面图，他们声称通过适当重排三脚架产生的每个袋中的三个顶点，可将平面图的队列数降至42。本注记指出了他们的队列布局算法中的一个漏洞：该算法所需的某一选择无法保证存在，因此已发表的证明并未确立所声称的42个队列的上界。
英文摘要
A queue layout of a graph consists of a linear order of the vertices and a partition of the edges into queues so that no two edges in a single queue are nested. The minimum number of queues needed in a queue layout of a graph is called its queue number. The planar product structure theorem states that every planar graph is a subgraph of the strong product of a graph of simple treewidth at most $3$, a clique $K_3$, and a path. Such a strong product admits a queue layout with $49$ queues (Wood, 2005), which implies that the queue number of planar graphs is at most $49$. Recently, Bekos, Gronemann, and Raftopoulou (Algorithmica, 2023) investigated how the general approach based on the product structure can be optimized for planar graphs. They claim that by appropriately reordering the three vertices in each bag arising from a tripod, it is possible to reduce the queue number of planar graphs to~$42$. In this note we highlight a gap in their queue layout algorithm: one of the choices required by the algorithm is not guaranteed to exist. Hence the claimed upper bound of $42$ queues is not established by the published proof.
