Distinct Distances in Graph Drawings - Carmi, Paz; Dujmović, Vida; Morin, Pat; Wood, David R. (The Electronic Journal of Combinatorics, 2008)
The 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 O(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 Δ increases the existential lower bound on the distance-number of Δ-regular graphs tends to Ω(n^0.864138).