Distinct Distances in Graph Drawings

TL;DR

This paper investigates the minimum number of distinct edge lengths needed to draw various classes of graphs, establishing bounds for graphs with bounded degree and treewidth, and constructing examples with high distance-numbers.

Contribution

It provides new bounds on the distance-number for graphs with bounded degree and treewidth, and constructs graphs with large distance-numbers, advancing understanding of geometric graph representations.

Findings

Graphs with bounded degree and treewidth have logarithmic distance-number.

Existence of graphs with maximum degree 5 and arbitrarily large distance-number.

Lower bounds on the distance-number of regular graphs grow as degree increases.

Abstract

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 $\Omega(n^{0.864138})$.