Extremal graphs for the odd prism

The Turán 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 Turán number of regular polyhedrons was widely studied in a series of works due to Simonovits. In this paper, we shall present the exact Turán 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 (2022) [55], 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 et al. (2022) [49], we will determine the exact value of $\mathrm{ex}(n,C_3^{\square })$ for every n instead of for sufficiently large n.