Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Upward Three-Dimensional Grid Drawings of Graphs   A \emph{three-dimensional grid drawing} of a graph is a placement of the
vertices at distinct points with integer coordinates, such that the straight
line segments representing the edges do not cross. Our aim is to produce
three-dimensional grid drawings with small bounding box volume. We prove that
every $n$-vertex graph with bounded degeneracy has a three-dimensional grid
drawing with $O(n^{3/2})$ volume. This is the broadest class of graphs admiting
such drawings. A three-dimensional grid drawing of a directed graph is
\emph{upward} if every arc points up in the z-direction. We prove that every
directed acyclic graph has an upward three-dimensional grid drawing with
$(n^3)$ volume, which is tight for the complete dag. The previous best upper
bound was $O(n^4)$. Our main result is that every $c$-colourable directed
acyclic graph ($c$ constant) has an upward three-dimensional grid drawing with
$O(n^2)$ volume. This result matches the bound in the undirected case, and
improves the best known bound from $O(n^3)$ for many classes of directed
acyclic graphs, including planar, series parallel, and outerplanar.

13
The categories of this paper are: quant-ph
15
The categories of this paper are: nucl-ex
3
The categories of this paper are: physics
36
The categories of this paper are: cmp-lg
0
The categories of this paper are: hep-ph
31
The categories of this paper are: ao-sci
26
The categories of this paper are: bayes-an
25
The categories of this paper are: patt-sol
29
The categories of this paper are: funct-an
11
The categories of this paper are: hep-lat
6
The categories of this paper are: astro-ph
18
The categories of this paper are: econ
9
The categories of this paper are: nlin
33
The categories of this paper are: plasm-ph
12
The categories of this paper are: nucl-th
8
The categories of this paper are: hep-ex