On the Tur\'an Number of Generalized Theta Graphs

Let ${\mathrm{\Theta }}_{k_1,\cdots ,k_{\mathrm{\ell }}}$ denote the generalized theta graph, which consists of $\mathrm{\ell }$ internally disjoint paths with lengths $k_1,\cdots ,k_{\mathrm{\ell }}$, connecting two fixed vertices. We estimate the corresponding extremal number $\text{ex}(n,{\mathrm{\Theta }}_{k_1,\cdots ,k_{\mathrm{\ell }}})$. When the lengths of all paths have the same parity and at most one path has length 1, $\text{ex}(n,{\mathrm{\Theta }}_{k_1,\cdots ,k_{\mathrm{\ell }}})$ is $O(n^{1+1\mathrm{/}{k}^{\ast }})$, where $2k^{\ast }$ is the length of the smallest cycle in ${\mathrm{\Theta }}_{k_1,\cdots ,k_{\mathrm{\ell }}}$. We also establish matching lower bound in the particular case of $\text{ex}(n,{\mathrm{\Theta }}_{3,5,5})$.