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

2026-08-05

AI Summary

This work challenges the claim by Bekos et al. that planar graphs admit queue layouts with at most 42 queues. Through a careful examination of their construction based on product structures of planar graphs, we identify for the first time a critical gap: the algorithm’s pivotal selection step lacks a guaranteed existence, thereby invalidating the original proof. By integrating tools from graph theory, combinatorial analysis, and queue layout theory, this study exposes the flaw in the prior argument and clarifies that the best currently valid upper bound on the queue number of planar graphs remains 49. Consequently, our findings correct a prevailing misconception in the field and reestablish the accurate boundary of known results concerning planar graph queue layouts.

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.

Problem

• queue layout
• planar graphs
• queue number
• product structure
• algorithmic gap

Innovation

• queue layout
• planar graphs
• queue number
• product structure
• algorithmic gap