k-path-edge-connectivity of the complete balanced bipartite graph

Abstract

Given a graph $G=(V,E)$ and a set $S\subseteq V\left(G\right)$ with $\left|S\right|\ge 2$, an S-path in G is a path that connects all vertices of S. Let ${\omega }_G\left(S\right)$ represent the maximum number of edge-disjoint S-paths in G. The k-path-edge-connectivity ${\omega }_k\left(G\right)$ of G is then defined as min$\left\{{\omega }_G\right(S):S\subseteq V(G\left)and\right|S|=k\}$, where $2\le k\le \left|V\right|$. Therefore, ${\omega }_2\left(G\right)$ is precisely the edge-connectivity $\lambda \left(G\right)$. In this paper, we focus on the k-path-edge-connectivity of the complete balanced bipartite graph $K_{n,n}$ for all $3\le k\le 2n$. We show that if $k=n$ or $k=n+1$, and n is odd, then ${\omega }_k\left(K_{n,n}\right)=\lfloor \frac{nk}{2(k-1)}\rfloor -1$; otherwise, ${\omega }_k\left(K_{n,n}\right)=\lfloor \frac{nk}{2(k-1)}\rfloor$.