关于广义Theta图的Turán数
On the Turán Number of Generalized Theta Graphs
AI总结：
该研究估计了广义theta图对应的Turán极值数，给出了所有路径奇偶性相同且至多一条长度为1时的上界，并在特定情形下建立了匹配的下界。
摘要：
设$\Theta_{k_1,\cdots,k_\ell}$表示广义theta图，它由$\ell$条长度为$k_1,\cdots,k_\ell$的内部不交路径连接两个固定顶点构成。我们估计了对应的极值数$\text{ex}(n,\Theta_{k_1,\cdots,k_\ell})$。当所有路径长度奇偶性相同且至多一条路径长度为1时，$\text{ex}(n,\Theta_{k_1,\cdots,k_\ell})$为$O(n^{1+1/k^\ast})$，其中$2k^\ast$是$\Theta_{k_1,\cdots,k_\ell}$中最小圈的长度。我们还在$\text{ex}(n,\Theta_{3,5,5})$的特定情形下建立了匹配的下界。
英文摘要：
Let $Θ_{k_1,\cdots,k_\ell}$ denote the generalized theta graph, which consists of $\ell$ internally disjoint paths with lengths $k_1,\cdots, k_{\ell}$, connecting two fixed vertices. We estimate the corresponding extremal number $\text{ex}(n,Θ_{k_1,\cdots,k_\ell})$. When the lengths of all paths have the same parity and at most one path has length 1, $\text{ex}(n,Θ_{k_1,\cdots,k_\ell})$ is $O(n^{1+1/k^\ast})$, where $2k^\ast$ is the length of the smallest cycle in $Θ_{k_1,\cdots,k_\ell}$. We also establish matching lower bound in the particular case of $\text{ex}(n,Θ_{3,5,5})$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
