The exact Turán number of generalized book graph $B_{r,k}$ in non-$r$-partite graphs

Given a graph $H,$ we say that a graph is $H\mathit{\text{-free}}$ if it does not contain $H$ as a subgraph. The Turán number $\text{ex}(n,H)$ of $H$ is the maximum number of edges in an $n$-vertex $H$-free graph, the set of all the corresponding extremal graphs is denoted by $\text{Ex}(n,H)$. The study of Turán number of graphs is a central topic in extremal graph theory. A graph is $\mathit{\text{color-critical}}$ if it contains an edge whose deletion reduces its chromatic number. Simonovits showed that if $H$ is a color-critical graph of chromatic number $r+1,$ then for sufficiently large $n,$ $\text{Ex}(n,H)=\{T_r(n)\},$ the $r$-partite Turán graph of order $n\mathrm{.}$ Given a color-critical graph $H$ with chromatic number $r+1,$ it is interesting to determine $H$-free non-$r$-partite graphs with maximum number of edges. For a graph $H$ with chromatic number $r+1,$ denote ${\text{ex}}_{r+1}(n,H)$ the maximum number of edges in non-$r$-partite $H$-free graphs of order $n,$ the set of all non-$r$-partite $H$-free graphs of order $n$ and size ${\text{ex}}_{r+1}(n,H)$ is denoted by ${\text{Ex}}_{r+1}(n,H)$. For $r\ge 3,k\ge 1,$ the generalized book graph $B_{r,k}$ is a graph obtained by joining every vertex of $K_r$ to every vertex of an independent set of size $k$. Note that $B_{r,k}$ is a color-critical graph of chromatic number $r+1.$ In this paper, based on the stability theory and local structure characterization, the exact value of ${\text{ex}}_{r+1}(n,B_{r,k})$ is determined and all the corresponding extremal graphs are identified, where $r\ge 3,k\ge 1$ and $n$ is sufficiently large.