Turán number of the odd-ballooning of complete bipartite graphs

Given a graph L $L$, the Tur & aacute;n number ex(n,L) $\text{ex}(n,L)$ is the maximum possible number of edges in an n $n$-vertex L $L$-free graph. The study of Tur & aacute;n number of graphs is a central topic in extremal graph theory. Although the celebrated Erd & odblac;s-Stone-Simonovits theorem gives the asymptotic value of ex(n,L) $\text{ex}(n,L)$ for nonbipartite L $L$, it is challenging in general to determine the exact value of ex(n,L) $\text{ex}(n,L)$ for chi(L)>= 3 $\chi (L)\ge 3$. The odd-ballooning of H $H$ is a graph such that each edge of H $H$ is replaced by an odd cycle and all new vertices of odd cycles are distinct. Here the length of odd cycles is not necessarily equal. The exact value of Tur & aacute;n number of the odd-ballooning of H $H$ is previously known for H $H$ being a cycle, a path, a tree with assumptions, and K2,3 ${K}_{2,3}$. In this paper, we manage to obtain the exact value of Tur & aacute;n number of the odd-ballooning of Ks,t ${K}_{s,t}$ with 2 <= s <= t $2\le s\le t$, where (s,t)is not an element of{(2,2),(2,3)} $(s,t)\notin \{(2,2),(2,3)\}$ and each odd cycle has length at least five.