Edge Partitions of Complete Geometric Graphs

Oswin Aichholzer, Johannes Obenaus, Joachim Orthaber, Rosna Paul, Patrick Schnider, Raphael Steiner, Tim Taubner, Birgit Vogtenhuber

Research output: Chapter in Book/Report/Conference proceedingConference paperpeer-review


In this paper, we disprove the long-standing conjecture that any complete geometric graph on 2n vertices can be partitioned into n plane spanning trees. Our construction is based on so-called bumpy wheel sets. We fully characterize which bumpy wheels can and in particular which cannot be partitioned into plane spanning trees (or even into arbitrary plane subgraphs). Furthermore, we show a sufficient condition for generalized wheels to not admit a partition into plane spanning trees, and give a complete characterization when they admit a partition into plane spanning double stars. Finally, we initiate the study of partitions into beyond planar subgraphs, namely into k-planar and k-quasi-planar subgraphs and obtain first bounds on the number of subgraphs required in this setting.

Original languageEnglish
Title of host publication38th International Symposium on Computational Geometry (SoCG 2022)
EditorsXavier Goaoc, Michael Kerber
Place of PublicationDagstuhl, Germany
PublisherSchloss Dagstuhl - Leibniz-Zentrum für Informatik
ISBN (Electronic)9783959772273
Publication statusPublished - 1 Jun 2022
Event38th International Symposium on Computational Geometry: SoCG 2022 - Berlin, Germany, Berlin, Germany
Duration: 7 Jun 202210 Jun 2022

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
ISSN (Print)1868-8969


Conference38th International Symposium on Computational Geometry
Abbreviated titleSoCG 2022
Internet address


  • complete geometric graph
  • edge partition
  • plane spanning tree
  • wheel set

ASJC Scopus subject areas

  • Software


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