Small cycles in the Pancake graph

Abstract

The Pancake graph is well known because of the open Pancake problem. It has the structure that any l –cycle, 6 ≤ l ≤ n ! , can be embedded in the Pancake graph P n , n ≥ 3 . Recently it was shown that there are exactly n ! / 6 independent 6 –cycles and n !( n − 3) distinct 7 –cycles in the graph. In this paper we characterize all distinct 8 –cycles by giving their canonical forms as products of generating elements. It is shown that there are exactly n !( n 3 + 12 n 2 − 103 n + 176) / 16 distinct 8 –cycles in P n , n ≥ 4 . A maximal set of independent 8 –cycles contains n ! / 8 of these.