## Abstract

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(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 Δ increases the existential lower bound on the distance-number of Δ-regular graphs tends to Ω(n^{0.864138}) .

Original language | English |
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Article number | R107 |

Journal | Electronic Journal of Combinatorics |

Volume | 15 |

Issue number | 1 |

DOIs | |

State | Published - 25 Aug 2008 |

Externally published | Yes |

## ASJC Scopus subject areas

- Theoretical Computer Science
- Geometry and Topology
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics
- Applied Mathematics