k-path-connectivity of the complete balanced tripartite graph Kn,n,n for n+1≤k≤2n−4

Given a graph $G=(V,E)$ and a set $S\subseteq V\left(G\right)$ of size at least 2, an $S$-path in $G$ is a path that connects all vertices of $S$. Two $S$-paths $P_1$ and $P_2$ are said to be internally disjoint if $E\left(P_1\right)\cap E\left(P_2\right)=0̸$ and $V\left(P_1\right)\cap V\left(P_2\right)=S$. Let ${\pi }_G\left(S\right)$ denote the maximum number of internally disjoint $S$-paths in $G$. The $k$-path-connectivity of $G$, denoted by ${\pi }_k\left(G\right)$, is then defined as ${\pi }_k\left(G\right)=\min \left\{{\pi }_G\left(S\right)\right|S\subseteq V\left(G\right)and\left|S\right|=k\}$, where $2\le k\le n$. Therefore, ${\pi }_2\left(G\right)$ is exactly the classical connectivity $\kappa \left(G\right)$, and ${\pi }_n\left(G\right)$ is exactly the maximum number of edge-disjoint Hamilton paths in $G$. It is established that for $3\le k\le n$, the $k$-path-connectivity ${\pi }_k$ of the complete balanced tripartite graph $K_{n,n,n}$ is $\lfloor \frac{2n}{k-1}\rfloor$. In this paper, we further derive that when $n+1\le k\le 2n-4$, ${\pi }_k\left(K_{n,n,n}\right)$ is $\lfloor \frac{3n-k}{2n-k-1}\rfloor$ for $n\ge 6$, and $\lfloor \frac{3n-k}{2n-k-1}\rfloor -1$ for $n=5$. As a corollary, we prove that if $c\le a+b+1$, then the complete tripartite graph $K_{a,b,c}$ contains at least $max\{\lfloor \frac{c}{2}\rfloor ,2\lfloor \frac{a-1}{2}\rfloor +1\}$ edge-disjoint Hamilton paths, where $a\le b\le c$.