LIPIcs.SoCG.2022.6.pdf
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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
-
Part of:
Volume:
38th International Symposium on Computational Geometry (SoCG 2022)
Series: Leibniz International Proceedings in Informatics (LIPIcs)
Conference: Symposium on Computational Geometry (SoCG) - License:
Creative Commons Attribution 4.0 International license
- Publication Date: 2022-06-01
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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 As Get 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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