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PreprintCC BY 4.0arXiv

Asymptotical results on the automorphism groups of canonical double cover of graphs

  • Bartłomiej Bychawski
  • Ademir Hujdurović
  • Đorđe Mitrović
  • Gabriel Verret
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Abstract

The canonical double cover BX of a graph X is the direct product X× K2. A graph X is stable if every automorphism of BX is induced by automorphisms of X and K2, and unstable otherwise. We prove several asymptotic results on stable and unstable graphs. In particular, we show that the automorphism group of the canonical double cover of almost every graph has order $2$, and consequently that almost all graphs are stable. We also prove that the dominant sources of instability are having twins and being disconnected, in that order, and that graphs arising from a known construction are the next step in this hierarchy.

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Cite

BibTeXRIS
BibTeX
@misc{bychawski2026asymptotical,
  title = {{Asymptotical results on the automorphism groups of canonical double cover of graphs}},
  author = {Bartłomiej Bychawski and Ademir Hujdurović and Đorđe Mitrović and Gabriel Verret},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.30373},
  eprint = {2610.30373},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.30373},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Asymptotical results on the automorphism groups of canonical double cover of graphs
AU  - Bartłomiej Bychawski
AU  - Ademir Hujdurović
AU  - Đorđe Mitrović
AU  - Gabriel Verret
PY  - 2026
DA  - 2026/10/21
PB  - arXiv
DO  - 10.48550/arxiv.2610.30373
UR  - https://arxiv.org/abs/2610.30373
AB  - The canonical double cover BX of a graph X is the direct product X× K_2. A graph X is stable if every automorphism of BX is induced by automorphisms of X and K_2, and unstable otherwise. We prove several asymptotic results on stable and unstable graphs. In particular, we show that the automorphism group of the canonical double cover of almost every graph has order $2$, and consequently that almost all graphs are stable. We also prove that the dominant sources of instability are having twins and being disconnected, in that order, and that graphs arising from a known construction are the next step in this hierarchy.
ER  -