Extremal permutations in routing cycles

Let G be a graph whose vertices are labeled 1, ... , n, and pi be a permutation on [n] := {1, 2, ... , n}. A pebble p(i) that is initially placed at the vertex i has destination pi(i) for each i is an element of [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) = 5, if rt(C-n, pi) = n-1, then pi = 23 ... n1 or its inverse. By a computer search, they showed that the conjecture holds for n \u3c 8. We prove in this paper that the conjecture holds for all even n \u3e= 6