非3部图中偶轮$W_{2k+2}$的精确Turán数
The exact Turán number of the even wheel $W_{2k+2}$ among non-$3$-partite graphs
AI总结：
本文确定了非3部图中偶轮$W_{2k+2}$的精确Turán数，并刻画了足够大$n$时对应的所有极值图。
AI中文摘要：
令$\text{ex}(n,H)$表示图$H$的Turán数；若图$H$存在边$e$使得$\text{χ}(H-e)<\text{χ}(H)$，则称$H$为色临界图。对于$\text{χ}(H)=r+1$的色临界图$H$，Simonovits的色临界边定理表明，存在$n_0(H)$，当$n\bg n_0(H)$时，$\text{ex}(n,H)=e(T_{n,r})$，且Turán图$T_{n,r}$是唯一极值图。设$W_{2k+2}$为将一个顶点连接到长度为$2k+1$的环所得的偶轮（$k\bg1$为整数），因$W_{2k+2}$是色临界图且$\text{χ}(W_{2k+2})=4$，足够大$n$时无$W_{2k+2}$图的唯一极值图为$T_{n,3}$，而$T_{n,3}$是3部图。本文确定了非3部图中$W_{2k+2}$的精确Turán数，并刻画了足够大$n$时的所有极值图。
英文摘要：
Let $\mathrm{ex}(n,H)$ denote the Turán number of $H$. A graph is color-critical if there exists an edge $e\in E(H)$ such that $χ(H-e)<χ(H)$. For a color-critical graph $H$ with $χ(H)=r+1$, Simonovits' chromatic critical edge theorem implies that there exists an $n_0(H)$ such that $\mathrm{ex}(n,H)=e(T_{n,r})$ and the Turán graph $T_{n,r}$ is the only extremal graph provided $n\geq n_0(H).$ Let $W_{2k+2}$ be the even wheel obtained by joining a vertex to a cycle of length $2k+1,$ where $k\geq1$ is an integer. Since $W_{2k+2}$ is color-critical and $χ(W_{2k+2})=4$, $T_{n,3}$ is the unique extremal graph for $W_{2k+2}$-free graphs of sufficiently large $n.$ Note that the extremal graph $T_{n,3}$ is 3-partite. In this paper, we determine the exact Turán number of $W_{2k+2}$ in non-$3$-partite graphs and characterize all extremal graphs provided $n$ is sufficiently large.
