Distinct distances in graph drawings
Electronic Journal of Combinatorics , Volume 15 - Issue 1
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(logn). 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(n0.864138).
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