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Planar Graphs of Bounded Degree have Constant Queue Number
A \emph{queue layout} of a graph consists of a \emph{linear order} of its
vertices and a partition of its edges into \emph{queues}, so that no two
independent edges of the same queue are nested. The \emph{queue number} of a
graph is the minimum number of queues required by any of its queue layouts. A
long-standing conjecture by Heath, Leighton and Rosenberg states that the queue
number of planar graphs is bounded. This conjecture has been partially settled
in the positive for several subfamilies of planar graphs (most of which have
bounded treewidth). In this paper, we make a further important step towards
settling this conjecture. We prove that planar graphs of bounded degree (which
may have unbounded treewidth) have bounded queue number.
A notable implication of this result is that every planar graph of bounded
degree admits a three-dimensional straight-line grid drawing in linear volume.
Further implications are that every planar graph of bounded degree has bounded
track number, and that every -planar graph (i.e., every graph that can be
drawn in the plane with at most crossings per edge) of bounded degree has
bounded queue number