Note on Path-Connectivity of Complete Bipartite Graphs

Abstract

For a graph $G=(V,E)$ and a set $S\subseteq V(G)$ of size at least $2$, a path in $G$ is said to be an $S$-path if it connects all vertices of $S$. Two $S$-paths $P_1$ and $P_2$ are said to be internally disjoint if $E(P_1)\cap E(P_2)=\varnothing$ and $V(P_1)\cap V(P_2)=S$. Let ${\pi }_G(S)$ denote the maximum number of internally disjoint $S$-paths in $G$. The $k$-path-connectivity ${\pi }_k(G)$ of $G$ is then defined as the minimum ${\pi }_G(S)$, where $S$ ranges over all $k$-subsets of $V(G)$. In [M. Hager, Path-connectivity in graphs, Discrete Math. 59(1986), 53--59], the $k$-path-connectivity of the complete bipartite graph $K_{a,b}$ was calculated, where $k\ge 2$. But, from his proof, only the case that $2\le k\le min\{a,b\}$ was considered. In this paper, we calculate the the situation that $min\{a,b\}+1\le k\le a+b$ and complete the result.