Dominant-Degree Conditions for Ramsey--Turán Factors of Non-Directed Cycle Orientations

@misc{zhou2026dominantdegree,
  title = {{Dominant-Degree Conditions for Ramsey--Turán Factors of Non-Directed Cycle Orientations}},
  author = {Jia Zhou and Yunshu Gao},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.00889},
  eprint = {2609.00889},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.00889},
}

TY  - UNPB
TI  - Dominant-Degree Conditions for Ramsey--Turán Factors of Non-Directed Cycle Orientations
AU  - Jia Zhou
AU  - Yunshu Gao
PY  - 2026
DA  - 2026/09/01
PB  - arXiv
DO  - 10.48550/arxiv.2609.00889
UR  - https://arxiv.org/abs/2609.00889
AB  - Let Cvec be a fixed orientation of the cycle C_ℓ, ℓ≥3, which is not directed. For an oriented graph D, let d_D^*(v):=max{d_D^+(v),d_D^-(v)}, and let sigore(D):=min{d_D^*(x)+d_D^*(y):x≠ y, xy,yx∉ A(D)}, with sigore(D)=∞ if the underlying graph of D is complete. We prove that, for every μ>0, there exist γ>0 and n_0 such that every n≥ n_0 with ℓ| n and every n-vertex oriented graph D satisfying α(D)≤γn and sigore(D)≥(3/4+μ)n contains a Cvec-factor. {Additionally, for every fixed s≥2 and every fixed real constant C, we construct arbitrarily large oriented graphs with sigore(D)≥ 3/4n+C that contain no C_2s^ad-factor. More precisely, C:=3/4α(D)−2 for s=2 and C:=1/4α(D)−3/2 for s≥3.} This paper develops a weighted reduction framework adapted to dominant degree condition, proves the absorption lemma via closed-cluster merging with even-walk, and derives almost-perfect tiling structures by virtue of Farkas-lemma-based fractional decomposition.
ER  -