%0 Journal Article
%T Distinct Distances in Graph Drawings
%A Paz Carmi
%A Vida Dujmovi£¿
%A Pat Morin
%A David R. Wood
%J Mathematics
%D 2008
%I arXiv
%X 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 $\Omega(n^{0.864138})$.
%U http://arxiv.org/abs/0804.3690v1