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