Extremal Permutations in Routing Cycles

Let $G$ be a graph whose vertices are labeled $1,\ldots,n$, and $\pi$ be a permutation on $[n]:=\{1,2,\ldots, n\}$. A pebble $p_i$ that is initially placed at the vertex $i$ has destination $\pi(i)$ for each $i\in [n]$. At each step, we choose a matching and swap the two pebbles on each of the edges. Let $rt(G, \pi)$, the routing number for $\pi$, be the minimum number of steps necessary for the pebbles to reach their destinations. Li, Lu and Yang proved that $rt(C_n, \pi)\le n-1$ for every permutation $\pi$ on the $n$-cycle $C_n$ and conjectured that for $n\geq 5$, if $rt(C_n, \pi) = n-1$, then $\pi = 23\cdots n1$ or its inverse. By a computer search, they showed that the conjecture holds for $n<8$. We prove in this paper that the conjecture holds for all even $n\ge 6$.