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

Let $\text{Cvec}$ be a fixed orientation of the cycle $C_{\mathrm{\ell }}$, $\mathrm{\ell }\ge 3$, which is not directed. For an oriented graph $D$, let $d_D^{\ast }(v):=\max \{d_D^{+}(v),d_D^{-}(v)\},$ and let $$\text{sigore}(D):=\min \{d_D^{\ast }(x)+d_D^{\ast }(y):x\mathrm{\ne }y,xy,yx\mathrm{\notin }A(D)\},$$ with $\text{sigore}(D)=\mathrm{\infty }$ if the underlying graph of $D$ is complete. We prove that, for every $\mu >0$, there exist $\gamma >0$ and $n_0$ such that every $n\ge n_0$ with $\mathrm{\ell }\mid n$ and every $n$-vertex oriented graph $D$ satisfying $$\alpha (D)\le \gamma n\text{ and }\text{sigore}(D)\ge (\frac{3}{4}+\mu )n$$ contains a $\text{Cvec}$-factor. {Additionally, for every fixed $s\ge 2$ and every fixed real constant $C$, we construct arbitrarily large oriented graphs with $\text{sigore}(D)\ge \frac{3}{4}n+C$ that contain no $C_{2s}^{\text{ad}}$-factor. More precisely, $C:=\frac{3}{4}\alpha (D)-2$ for $s=2$ and $C:=\frac{1}{4}\alpha (D)-\frac{3}{2}$ for $s\ge 3$.} 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.