Cyclic Hamilton Cycle Decompositions of Carousel Tournaments of Order pq

@misc{liao2026cyclic,
  title = {{Cyclic Hamilton Cycle Decompositions of Carousel Tournaments of Order $pq$}},
  author = {Hongci Liao and Yongju Peng and Guang Li and Yingbin Ma},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.07102},
  eprint = {2610.07102},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.07102},
}

TY  - UNPB
TI  - Cyclic Hamilton Cycle Decompositions of Carousel Tournaments of Order pq
AU  - Hongci Liao
AU  - Yongju Peng
AU  - Guang Li
AU  - Yingbin Ma
PY  - 2026
DA  - 2026/10/05
PB  - arXiv
DO  - 10.48550/arxiv.2610.07102
UR  - https://arxiv.org/abs/2610.07102
AB  - Kelly's conjecture asks whether every regular tournament admits a Hamilton cycle decomposition. Motivated by its symmetry-preserving extension, we study cyclic Hamilton decompositions of carousel tournaments. For an odd integer n, let T_n=Cay( ℤ_n,{1,2,…,(n−1)/2} ) be the carousel tournament. We ask whether T_n has a Hamilton decomposition invariant under translation by $1$. Although the answer is immediate when n is prime, composite orders introduce a genuine obstruction: nonunit differences generate short cycles rather than Hamilton cycles. We resolve a general composite-order family by proving that, whenever n=pq for primes 7≤ p<q, the tournament T_n admits a cyclic Hamilton cycle decomposition. The proof combines Hamiltonian difference sequences over prime fields with a matching argument that constructs two base paths with disjoint difference sets in ℤ_pq. Thus our result gives an infinite family supporting the cyclic, symmetry-preserving extension of the Hamilton decomposition problem for regular tournaments.
ER  -