无向环定向的Ramsey–Turán因子的控制度条件

中文摘要 AI 辅助

设$\u27C8$是环$C_\u2113$（$\u2113\b3$）的一个固定非有向定向。对于有向图$D$，定义$d_D^*(v):=\u006dx{d_D^+(v),d_D^-(v)}$，并定义$\u03c3ore(D):=\u006dn{d_D^*(x)+d_D^*(y):x\neq y,\u3000xy,yx\notin A(D)}$，若$D$的基础图是完全图则$\u03c3ore(D)=\u221e$。我们证明，对任意$\u03bc>0$，存在$\u03b3>0$和$n_0$，使得所有满足$\u2113\bmid n$的$n\b n_0$，以及所有满足$\u03b1(D)\u2264\u03b3n$且$\u03c3ore(D)\b(\frac{3}{4}+\u03bc)n$的$n$顶点有向图$D$，都包含一个$\u27C8$-因子。此外，对任意固定$s\b2$和任意固定实常数$C$，我们构造了任意大的满足$\u03c3ore(D)\b \frac{3}{4}n+C$但不含$C_{2s}^{ad}$-因子的有向图。更精确地，$s=2$时$C:=\frac{3}{4}\u03b1(D)-2$，$s\b3$时$C:=\frac{1}{4}\u03b1(D)-\frac{3}{2}$。本文建立了适配控制度条件的加权约简框架，通过带偶步长的闭簇合并证明吸收引理，并借助基于Farkas引理的分数分解得到近乎完美的平铺结构。

英文摘要

Let $\Cvec$ be a fixed orientation of the cycle $C_\ell$, $\ell\ge3$, which is not directed. For an oriented graph $D$, let $d_D^*(v):=\max\{d_D^+(v),d_D^-(v)\},$ 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 $μ>0$, there exist $γ>0$ 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 $C_{2s}^{\ad}$-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.

发表机构

• School of Mathematics and Statistics, Ningxia University（宁夏大学数学与统计学院）

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