Johnson graphs are panconnected
For any given $n,m \in \mathbb{N}$ with $ m < n $, the Johnson graph $J(n,m)$ is defined as the graph whose vertex set is $V=\{v\mid v\subseteq [n]=\{1,...,n\}, |v|=m\}$, where two vertices $v$,$w$ are adjacent if and only if $|v\cap w|=m-1$. A graph $G$ of order $n > 2$ is panconnected if for every two vertices $u$ and $v$, there is a $u$-$v$ path of length $l$ for every integer $l$ with $d(u,v) \leq l \leq n-1$. In this paper, we prove that the Johnson graph $J(n,m)$ is a panconnected graph.
math.GR↗