On the Tur\'an Number of Generalized Theta Graphs

Let $\Theta_{k_1,\cdots,k_\ell}$ 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 $\text{ex}(n,\Theta_{k_1,\cdots,k_\ell})$. When the lengths of all paths have the same parity and at most one path has length 1, $\text{ex}(n,\Theta_{k_1,\cdots,k_\ell})$ is $O(n^{1+1/k^\ast})$, where $2k^\ast$ is the length of the smallest cycle in $\Theta_{k_1,\cdots,k_\ell}$. We also establish matching lower bound in the particular case of $\text{ex}(n,\Theta_{3,5,5})$.