Extremal graphs for the odd prism

The Tur\'an number $\mathrm{ex}(n,H)$ of a graph $H$ is the maximum number of edges in an $n$-vertex graph which does not contain $H$ as a subgraph. The Tur\'{a}n number of regular polyhedrons was widely studied in a series of works due to Simonovits. In this paper, we shall present the exact Tur\'{a}n number of the prism $C_{2k+1}^{\square}$, which is defined as the Cartesian product of an odd cycle $C_{2k+1}$ and an edge $K_2$. Applying a deep theorem of Simonovits and a stability result of Yuan [European J. Combin. 104 (2022)], we shall determine the exact value of $\mathrm{ex}(n,C_{2k+1}^{\square})$ for every $k\ge 1$ and sufficiently large $n$, and we also characterize the extremal graphs. Moreover, in the case of $k=1$, motivated by a recent result of Xiao, Katona, Xiao and Zamora [Discrete Appl. Math. 307 (2022)], we will determine the exact value of $\mathrm{ex}(n,C_{3}^{\square} )$ for every $n$ instead of for sufficiently large $n$.