Extremal Permutations in Routing Cycles

Let G be a graph whose vertices are labeled 1, . . . , n, and π be a permutation on [n] := {1, 2, . . . , n}. A pebble pi that is initially placed at the vertex i has destination π(i) for each i ∈ [n]. At each step, we choose a matching and swap the two pebbles on each of the edges. Let rt(G, π), the routing number for π, be the minimum number of steps necessary for the pebbles to reach their destinations. Li, Lu and Yang proved that rt(Cn, π) 6 n− 1 for every permutation π on the n-cycle Cn and conjectured that for n > 5, if rt(Cn, π) = n−1, then π = 23 · · ·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 > 6.