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

Abstract: Let $\Cvec$ be a fixed orientation of the cycle $C_\ell$, $\ell\ge3$, which is not directed. For an oriented graph $D$, let $d_D<sup>*(v):=\max{d_D<sup>+(v),d_D<sup>-(v)},$ and let [ \sigore(D):=\min\bigl{d_D^(x)+d_D^(y):x\ne y,\ xy,yx\notin A(D)\bigr}, ] with $\sigore(D)=\infty$ if the underlying graph of $D$ is complete. We prove that, for every $μ&gt;0$, there exist $γ&gt;0$ and $n_0$ such that every $n\ge n_0$ with $\ell\mid n$ and every $n$-vertex oriented graph $D$ satisfying [ α(D)\leγn \text{ and } {\sigore(D)\ge\left(\frac34+μ\right)n} ] contains a $\Cvec$-factor. {Additionally, for every fixed $s\ge2$ and every fixed real constant $C$, we construct arbitrarily large oriented graphs with $\sigore(D)\ge \frac34n+C$ that contain no $C_{2s}<sup>{\ad}$-factor. More precisely, $C:=\frac34α(D)-2$ for $s=2$ and $C:=\frac14α(D)-\frac32$ for $s\ge3$.} 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.