The generalized connectivity of complete bipartite graphs

Let $G$ be a nontrivial connected graph of order $n$, and $k$ an integer with $2\le k\le n$. For a set $S$ of $k$ vertices of $G$, let $\kappa (S)$ denote the maximum number $\mathrm{\ell }$ of edge-disjoint trees $T_1,T_2,\mathrm{.}\mathrm{.}\mathrm{.},T_{\mathrm{\ell }}$ in $G$ such that $V(T_i)\cap V(T_j)=S$ for every pair $i,j$ of distinct integers with $1\le i,j\le \mathrm{\ell }$. Chartrand et al. generalized the concept of connectivity as follows: The $k$-$connectivity$, denoted by ${\kappa }_k(G)$, of $G$ is defined by ${\kappa }_k(G)=$min$\{\kappa (S)\}$, where the minimum is taken over all $k$-subsets $S$ of $V(G)$. Thus ${\kappa }_2(G)=\kappa (G)$, where $\kappa (G)$ is the connectivity of $G$. Moreover, ${\kappa }_n(G)$ is the maximum number of edge-disjoint spanning trees of $G$. This paper mainly focus on the $k$-connectivity of complete bipartite graphs $K_{a,b}$. First, we obtain the number of edge-disjoint spanning trees of $K_{a,b}$, which is $\lfloor \frac{ab}{a+b-1}\rfloor$, and specifically give the $\lfloor \frac{ab}{a+b-1}\rfloor$ edge-disjoint spanning trees. Then based on this result, we get the $k$-connectivity of $K_{a,b}$ for all $2\leq k \leq a+b$\mathrm{.}Namely,if$k>b-a+2$and$a-b+k$isoddthen$\kappa_{k}(K_{a,b})=\frac{a+b-k+1}{2}+\lfloor\frac{(a-b+k-1)(b-a+k-1)}{4(k-1)}\rfloor,$if$k>b-a+2$and$a-b+k$iseventhen$\kappa_{k}(K_{a,b})=\frac{a+b-k}{2}+\lfloor\frac{(a-b+k)(b-a+k)}{4(k-1)}\rfloor,$andif$k\leq b-a+2$then$\kappa_{k}(K_{a,b})=a. $