4-Arc-Pancyclicity of Regular Multipartite Tournaments

@misc{xia20264arcpancyclicity,
  title = {{4-Arc-Pancyclicity of Regular Multipartite Tournaments}},
  author = {Weihao Xia},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.12372},
  eprint = {2609.12372},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.12372},
}

TY  - UNPB
TI  - 4-Arc-Pancyclicity of Regular Multipartite Tournaments
AU  - Weihao Xia
PY  - 2026
DA  - 2026/09/11
PB  - arXiv
DO  - 10.48550/arxiv.2609.12372
UR  - https://arxiv.org/abs/2609.12372
AB  - A multipartite tournament is an orientation of a complete multipartite graph. We prove that every r-regular c-partite tournament with common partite-set cardinality α is $4$-arc-pancyclic whenever c≥93; that is, every arc belongs to a cycle of each length from $4$ to cα. This confirms the conjecture of Zhou and Zhang for all sufficiently large c and provides a multipartite analog of Alspach's arc-pancyclicity theorem. Moreover, we also give a construction to show that 4-arc-pancyclic is the best possible. Next, we prove that every arc belongs to at least cα−α−1 cycles of pairwise distinct lengths when c≥7 and α≥2. For regular $3$-partite tournaments with common partite-set cardinality α≥2, we obtain the sharp lower bound α, settling the remaining case of a conjecture of Xia, Cai, Guo, and Wang.
ER  -