Cayley Tournaments Simultaneously Critical for the Clique and Dichromatic Numbers

@misc{chen2026cayley,
  title = {{Cayley Tournaments Simultaneously Critical for the Clique and Dichromatic Numbers}},
  author = {Guantao Chen and Shengze Wang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.08658},
  eprint = {2609.08658},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.08658},
}

TY  - UNPB
TI  - Cayley Tournaments Simultaneously Critical for the Clique and Dichromatic Numbers
AU  - Guantao Chen
AU  - Shengze Wang
PY  - 2026
DA  - 2026/09/08
PB  - arXiv
DO  - 10.48550/arxiv.2609.08658
UR  - https://arxiv.org/abs/2609.08658
AB  - For a tournament T, let ω(T) be the minimum clique number among the backedge graphs of T, and let χ(T) be its dichromatic number. We give a template-lifting construction. It turns a k-template into a regular, vertex-transitive Cayley tournament that is simultaneously (k+1)-ω-critical and (k+1)-χ-critical. The output is also a (k+1)-template. Iterating the construction, we prove that for every k≥3, there is a positive even integer m_k with the following property. Every N>1 with N≡1pmodm_k is the order of a regular, vertex-transitive Cayley tournament that is simultaneously k-ω-critical and k-χ-critical. This proves a conjecture of Aboulker, Aubian, Charbit, and Lopes and gives a negative answer to their bounded-certificate question when the hypothesis is ω(T)≥ k. We also find the clique number of a cyclic substitution when each block satisfies ω=χ. We then describe exactly when this substitution is ω-critical if the blocks are χ-critical and satisfy ω=χ.
ER  -