Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Distinct Distances in Graph Drawings   The \emph{distance-number} of a graph $G$ is the minimum number of distinct
edge-lengths over all straight-line drawings of $G$ in the plane. This
definition generalises many well-known concepts in combinatorial geometry. We
consider the distance-number of trees, graphs with no $K^-_4$-minor, complete
bipartite graphs, complete graphs, and cartesian products. Our main results
concern the distance-number of graphs with bounded degree. We prove that
$n$-vertex graphs with bounded maximum degree and bounded treewidth have
distance-number in $\mathcal{O}(\log n)$. To conclude such a logarithmic upper
bound, both the degree and the treewidth need to be bounded. In particular, we
construct graphs with treewidth 2 and polynomial distance-number. Similarly, we
prove that there exist graphs with maximum degree 5 and arbitrarily large
distance-number. Moreover, as $\Delta$ increases the existential lower bound on
the distance-number of $\Delta$-regular graphs tends to $\Omega(n^{0.864138})$.

0
The categories of this paper are: hep-ph
30
The categories of this paper are: q-alg
16
The categories of this paper are: stat
18
The categories of this paper are: econ
28
The categories of this paper are: alg-geom
9
The categories of this paper are: nlin
21
The categories of this paper are: adap-org
11
The categories of this paper are: hep-lat
7
The categories of this paper are: hep-th
1
The categories of this paper are: cs
34
The categories of this paper are: chem-ph
12
The categories of this paper are: nucl-th
25
The categories of this paper are: patt-sol
35
The categories of this paper are: mtrl-th
3
The categories of this paper are: physics
6
The categories of this paper are: astro-ph