A Gap in the 42-Queue Layout Algorithm for Planar Graphs

Abstract

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.