On the number of Fk,4-saturating edges

For a graph F, let G be an F-free graph, a non-edge e of G is an F-saturating edge if $G+e$ contains a copy of F. Graph $F_{k,r}$ consists of k cliques $K_r$ intersecting in exactly one common vertex. Denote by $f_F(n,m)$ the minimum number of F-saturating edges of F-free graphs on n vertices with m edges and $f_F^{⁎}(n,m)$, where $m\le ex(n,F)$, the minimum number of F-saturating edges of F-free graphs on n vertices with m edges obtained by deleting edges from the extremal graph attaining $ex(n,F)$. In this paper, we study the number of $F_{k,4}$-saturating edges in $F_{k,4}$-free graphs on n vertices with $ex(n,F_{k-1,4})+1$ edges. We give the upper bounds on $f_{F_{k,4}}(n,ex(n,F_{k-1,4})+1)$ and get the value of $f_{F_{2,4}}(n,ex(n,F_{1,4})+1)$. Moreover, we characterize the extremal graphs attaining $ex(n,F_{k,4}$) with odd $k\ge 3$ and prove $f_{F_{k,4}}^{⁎}(n,ex(n,F_{k-1,4})+1)=\lfloor \frac{n}{3}\rfloor -k$ for odd $k\ge 3$.