Edge Partitions of Complete Geometric Graphs (Part 1)
Combinatorics
2021-12-20 v2 Computational Geometry
Abstract
In this paper, we disprove the long-standing conjecture that any complete geometric graph on vertices can be partitioned into plane spanning trees. Our construction is based on so-called bumpy wheel sets. We fully characterize which bumpy wheels can and in particular which \emph{cannot} be partitioned into plane spanning trees (or even into arbitrary plane \emph{subgraphs}), including a complete description of all possible partitions (into plane spanning trees). Furthermore, we show a sufficient condition for \emph{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.
Keywords
Cite
@article{arxiv.2108.05159,
title = {Edge Partitions of Complete Geometric Graphs (Part 1)},
author = {Johannes Obenaus and Joachim Orthaber},
journal= {arXiv preprint arXiv:2108.05159},
year = {2021}
}