On sparse universally rigid graphs in ℝ

@misc{gabsdil2026sparse,
  title = {{On sparse universally rigid graphs in $\mathbbR$}},
  author = {Ole Gabsdil and Ezra Hirschi and Julien Portier},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.41581},
  eprint = {2610.41581},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.41581},
  note = {CC BY-NC-ND 4.0},
}

TY  - UNPB
TI  - On sparse universally rigid graphs in ℝ
AU  - Ole Gabsdil
AU  - Ezra Hirschi
AU  - Julien Portier
PY  - 2026
DA  - 2026/10/30
PB  - arXiv
DO  - 10.48550/arxiv.2610.41581
UR  - https://arxiv.org/abs/2610.41581
AB  - A graph is said to be 1-universally rigid if, for any generic realization of its vertices on the real line, the realization cannot be deformed out of the line in any higher-dimensional Euclidean space while preserving all edge lengths. In this paper, we investigate the sparsity of 1-universally rigid graphs in different manners. First, we show that an Erdős-Rényi random graph G(n,p) is 1-universally rigid with high probability whenever p ≥ Clog n · loglog n/n for some sufficiently large constant C>0. These techniques also imply the existence of 1-universally rigid graphs of arbitrarily large girth, which resolves a problem suggested by Dantas e Moura, Jordán, and Silverman in a non-constructive manner. We also study how sparse a 1-universally rigid graph can be, proving that there exists an absolute constant ε>0 such that every sufficiently large 1-universally rigid graph on n vertices has at least (3/2+ε)n edges. This breaks the natural 3/2 barrier arising from degree-2 extensions and makes progress towards a question of Jordán and Nguyen.
ER  -