Cycles in the burnt pancake graph

Abstract

The pancake graph Pn is the Cayley graph of the symmetric group Sn on n elements generated by prefix reversals. Pn has been shown to have properties that makes it a useful network scheme for parallel processors. For example, it is (n − 1)-regular, vertex-transitive, and one can embed cycles in it of length ℓ with 6 ≤ ℓ ≤ n!. The burnt pancake graph BPn, which is the Cayley graph of the group of signed permutations Bn using prefix reversals as generators, has similar properties. Indeed, BPn is n-regular and vertextransitive. In this paper, we show that BPn has every cycle of length ℓ with 8 ≤ ℓ ≤ 2 n n!. The proof given is a constructive one that utilizes the recursive structure of BPn.We also present a complete characterization of all the 8-cycles in BPn for n ≥ 2, which are the smallest cycles embeddable in BPn, by presenting their canonical forms as products of the prefix reversal generators.

An Algorithm to Enumerate Grid Signed Permutation Classes

“…But there are some permutations we are still double counting. Note that 12 [ 1,3 ], 12 [ 2,2 ], and 12 [ 3,1 ] all produce 1234. Definition 3.2 (Compact signed permutations).…”

The Construction of Multiple Independent Spanning Trees on Burnt Pancake Networks

“…Let π n k (u 1 , u 2 ) denote the vertex subset ⊂ π n k , whose first and second elements are u 1 and u 2 , respectively. For instance, there exist two vertices u (2, 3, 4, 1) and u (3, 2, 4, 1) ∈ π 4 1 , where vertex u ∈ π 4 1 (2, 3) and vertex u ∈ π 4 1 (3,2). Let u(i), where 1 ≤ i ≤ n, denote the i th element of vertex u.…”

“…edges. For more research results related to burnt pancake graphs, we refer to [2], [10], [15], [29], [30]. For instance, Figure 1 and Figure 2 depict 2-dimensional and 3-dimensional burnt pancake networks, respectively.…”

“…Let π n denote the set of all permutations of n distinct elements from 1 to n, where all elements are signed or unsigned. One example of π n is π 2…”

“…2 a) If (k = the smaller one of u(1), u(2) and u(1) is signed) || (k = the larger one of u(1), u(2) and u(1) is unsigned),table[u, k] = u 1 . b) If (k = the smaller one of u(1), u(2) and u(1) is unsigned) || (k = the larger one of u(1), u(2) and u(1) is signed), table[u, k]= u 2 .…”