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Edge Partitions of Complete Geometric Graphs

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



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Author Details

Oswin Aichholzer
  • Institute of Software Technology, Technische Universität Graz, Austria
Johannes Obenaus
  • Department of Computer Science, Freie Universität Berlin, Germany
Joachim Orthaber
  • Institute of Software Technology, Technische Universität Graz, Austria
Rosna Paul
  • Institute of Software Technology, Technische Universität Graz, Austria
Patrick Schnider
  • Department of Mathematical Sciences, University of Copenhagen, Denmark
Raphael Steiner
  • Department of Computer Science, ETH Zürich, Switzerland
Tim Taubner
  • Department of Computer Science, ETH Zürich, Switzerland
Birgit Vogtenhuber
  • Institute of Software Technology, Technische Universität Graz, Austria

Acknowledgements

Research on this work has been initiated in March 2021, at the 5^{th} research workshop of the collaborative D-A-CH project Arrangements and Drawings, which was funded by the DFG, the FWF, and the SNF. We thank the organizers and all participants for fruitful discussions. Further, we thank the anonymous reviewers for their insightful comments and suggestions.

Cite AsGet BibTex

Oswin Aichholzer, Johannes Obenaus, Joachim Orthaber, Rosna Paul, Patrick Schnider, Raphael Steiner, Tim Taubner, and Birgit Vogtenhuber. Edge Partitions of Complete Geometric Graphs. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 6:1-6:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.SoCG.2022.6

Abstract

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.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatorics
  • Mathematics of computing → Graph theory
Keywords
  • edge partition
  • complete geometric graph
  • plane spanning tree
  • wheel set

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References

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