About an extremal problem of bigraphic pairs with a realization containing ${\displaystyle K_{s,t}}$

Let ${\displaystyle \pi =(f_1,...,f_m;g_1,...,g_n)}$, where ${\displaystyle f_1,...,f_m}$ and ${\displaystyle g_1,...,g_n}$ are two non-increasing sequences of nonnegative integers. The pair ${\displaystyle \pi =(f_1,...,f_m;}$ ${\displaystyle g_1,...,g_n)}$ is said to be a bigraphic pair if there is a simple bipartite graph ${\displaystyle G=(X\cup Y,E)}$ such that ${\displaystyle f_1,...,f_m}$ and ${\displaystyle g_1,...,g_n}$ are the degrees of the vertices in ${\displaystyle X}$ and ${\displaystyle Y}$, respectively. In this case, ${\displaystyle G}$ is referred to as a realization of ${\displaystyle \pi }$. We say that ${\displaystyle \pi }$ is a potentially ${\displaystyle K_{s,t}}$-bigraphic pair if some realization of ${\displaystyle \pi }$ contains ${\displaystyle K_{s,t}}$ (with ${\displaystyle s}$ vertices in the part of size ${\displaystyle m}$ and ${\displaystyle t}$ in the part of size ${\displaystyle n}$). Ferrara et al. [Potentially ${\displaystyle H}$-bigraphic sequences, Discuss. Math. Graph Theory 29 (2009) 583–596] defined ${\displaystyle \sigma (K_{s,t},m,n)}$ to be the minimum integer ${\displaystyle k}$ such that every bigraphic pair ${\displaystyle \pi =(f_1,...,f_m;g_1,...,g_n)}$ with ${\displaystyle \sigma \left(\pi \right)=f_1+\cdots +f_m\ge k}$ is potentially ${\displaystyle K_{s,t}}$-bigraphic. They determined ${\displaystyle \sigma (K_{s,t},m,n)}$ for ${\displaystyle n\ge m\ge 9s^4{t}^4}$. In this paper, we first give a procedure and two sufficient conditions to determine if ${\displaystyle \pi }$ is a potentially ${\displaystyle K_{s,t}}$-bigraphic pair. Then, we determine ${\displaystyle \sigma (K_{s,t},m,n)}$ for ${\displaystyle n\ge m\ge s}$ and ${\displaystyle n\ge (s+1)t^2-(2s-1)t+s-1}$. This provides a solution to a problem due to Ferrara et al.