Mutually Independent Hamiltonian Cycle of Burnt Pancake Graphs

Let ( ) , G V E = be a graph of order n. A Hamiltonian cycle of G is a cycle contains every vertex in G. Two Hamiltonian cycles 1 1 2 3 1 , , ,... , n C u u u u u = and C2 = 1 2 3 1 , , ,... , n v v v v v of G are independent if 1 1 u v = and i i u v ≠ for 2 i n ≤ ≤ . A set of Hamiltonian cycles { } 1 2 3 , , ,... k C C C C of G is mutually independent if its elements are pairwise independent. The mutually independent hamiltonicity ( ) IHC G of a graph G is the maximum integer k such that for any vertex u of G there are k mutually independent Hamiltonian cycles of G starting at u. For the n-dimensional burnt pancake graph n B , this paper proved that ( ) 2 1 IHC B = and ( ) n IHC B n = 3 for n ≥