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1-Planar Unit Distance Graphs

Authors Panna Gehér, Géza Tóth

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Author Details

Panna Gehér
  • Eötvös Loránd University, Budapest, Hungary
Géza Tóth
  • Alfréd Rényi Institute of Mathematics, Budapest, Hungary

Funding

  • Gehér, Panna: Supported by Lendület Programme of the Hungarian Academy of Sciences, LP2021-1/2021, ERC Advanced Grant "GeoScape" No. 882971 and by the National Research, Development and Innovation Office, NKFIH, K-131529.
  • Tóth, Géza: Supported by ERC Advanced Grant "GeoScape" No. 882971 and by the National Research, Development and Innovation Office, NKFIH, K-131529.

Cite AsGet BibTex

Panna Gehér and Géza Tóth. 1-Planar Unit Distance Graphs. In 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 320, pp. 6:1-6:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.GD.2024.6

Abstract

A matchstick graph is a plane graph with edges drawn as unit distance line segments. This class of graphs was introduced by Harborth who conjectured that a matchstick graph on n vertices can have at most ⌊3n-√{12n-3}⌋ edges. Recently his conjecture was settled by Lavollée and Swanepoel. In this paper we consider 1-planar unit distance graphs. We say that a graph is a 1-planar unit distance graph if it can be drawn in the plane such that all edges are drawn as unit distance line segments while each of them are involved in at most one crossing. We show that such graphs on n vertices can have at most 3n-∜{n}/10 edges.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Discrete mathematics
Keywords
  • unit distance graph
  • 1-planar
  • matchstick graph

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