Some results on Archdeacon's conjecture for rotation systems

@misc{chikkatur2026some,
  title = {{Some results on Archdeacon's conjecture for rotation systems}},
  author = {Arahat Chikkatur and Ji Zeng},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.11599},
  eprint = {2609.11599},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.11599},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - Some results on Archdeacon's conjecture for rotation systems
AU  - Arahat Chikkatur
AU  - Ji Zeng
PY  - 2026
DA  - 2026/09/10
PB  - arXiv
DO  - 10.48550/arxiv.2609.11599
UR  - https://arxiv.org/abs/2609.11599
AB  - A rotation system on n elements assigns to each element a cyclic order of the other n−1 elements. A four-element subset is non-planar if its induced rotation system cannot be realized by a crossing-free drawing of K_4. As a combinatorial strengthening of Hill's conjecture on the crossing number of the complete graph, Archdeacon conjectured that every rotation system on n elements has at least H(n)=1/4 ⌊n/2⌋ ⌊(n−1)/2⌋ ⌊(n−2)/2⌋ ⌊(n−3)/2⌋ non-planar four-element subsets. We computationally verify Archdeacon's conjecture for n≤ 10 and show that every extremal rotation system in these orders is realizable by a simple drawing. With computer assistance, we prove that every rotation system on n elements has at least $(8/9 - o(1)) H(n)$ non-planar four-element subsets. We also present a proof by hand for a weaker lower bound of $(2/3-o(1)) H(n)$. Finally, extending recent work of Felsner on antipodal pairs in drawings, we show that Archdeacon's conjecture holds for antipodally shellable rotation systems.
ER  -