图的有界VC色阈值
The Bounded-VC chromatic thresholds of graphs
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中文总结 AI 辅助
本文研究图的有界VC色阈值，针对$\u03c7(H)\u22653$的图$H$，基于其分解族是否含森林，明确了该阈值的取值，拓展了色阈值的相关研究。
中文摘要 AI 辅助
对于图$H$，色阈值$\u03b4_\u03c7(H)$是满足以下条件的最小$c>0$：每个$n$顶点、最小度至少为$cn$的不含$H$的图的色数，被仅依赖于$H$和$c$的常数界定。Allen、Böttcher、Griffiths、Kohayakawa和Morris证明，若$\u03c7(H)=r\u22653$，则$\u03b4_\u03c7(H)\u2208\left\{\frac{r-3}{r-2},\frac{2r-5}{2r-3},\frac{r-2}{r-1}\right\}$。Liu、Shangguan、Skokan和Xu通过限制宿主图具有有界VC维，引入了有界VC色阈值$\text{VC}(H)$。本文确定了$\u03c7(H)\u22653$的图$H$的该参数：设$\u039c(H)$为$r$色图$H$的分解族，则$\text{VC}(H)=\begin{cases}\dfrac{r-3}{r-2},&\text{若$\u039c(H)$包含森林},\\\\[4pt]\dfrac{r-2}{r-1},&\text{否则}.\end{cases}$
英文摘要
For a graph $H$, the chromatic threshold $δ_χ(H)$ is the infimum of $c>0$ such that the chromatic number of every $n$-vertex $H$-free graph with minimum degree at least $cn$ is bounded by a constant depending only on $H$ and $c$. Allen, Böttcher, Griffiths, Kohayakawa, and Morris proved that if $χ(H)=r\geq 3$, then $δ_χ(H)\in\{\frac{r-3}{r-2}, \frac{2r-5}{2r-3}, \frac{r-2}{r-1}\}$. Liu, Shangguan, Skokan, and Xu introduced the bounded-VC chromatic threshold $\text{VC}(H)$ by restricting the host graphs to have bounded VC-dimension. We determine this parameter for graph $H$ with $χ(H)\ge 3$. More precisely, let $\mathcal{M}(H)$ be the decomposition family of an $r$-chromatic graph $H$, then \[ \text{VC}(H)= \begin{cases} \dfrac{r-3}{r-2},&\text{if $\mathcal{M}(H)$ contains a forest},\\[4pt] \dfrac{r-2}{r-1},&\text{otherwise}. \end{cases} \]
