Note on Path-Connectivity of Complete Bipartite Graphs

TL;DR

This paper extends the calculation of $k$-path-connectivity in complete bipartite graphs to cases where $k$ exceeds the smaller part size, completing the understanding of path connectivity for all $k$.

Contribution

It completes the determination of $k$-path-connectivity for complete bipartite graphs for all values of $k$, including those greater than the smaller part size.

Findings

Calculated $k$-path-connectivity for $minigrace a,bigrace +1 \,\leq k \leq a+b$

Extended previous results to cover all $k$ in complete bipartite graphs

Provided a complete characterization of path connectivity for all $k$

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 $E(P_1)\cap E(P_2)=\emptyset$ 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 $k\geq 2$. But, from his proof, only the case that $2\leq k\leq min\{a,b\}$ was considered. In this paper, we calculate the the situation that $min\{a,b\}+1\leq k\leq a+b$ and complete the result.