Mutually Independent Hamiltonianicity of Pancake Graphs and Star Graphs

A hamiltonian cycle C of a graph G is described as langu 1 , u 2 ,..., u n(G) , u 1 rang to emphasize the order of vertices in C. Thus, u 1 is the start vertex and u i is the i-th vertex in C. Two hamiltonian cycles of G start at a vertex x, C 1 = langu 1 , u 2 ,..., u n(G) , u 1 rang and C 2 = langv 1 , v 2 ,..., v n(G) , v 1 rang, are independent if x = u 1 = v 1 and u 1 ne v i for every i, 2 les i les n(G). A set of hamiltonian cycles {C 1 , C 2 ,..., C k } of G are mutually independent if any two different hamiltonian cycles are independent. The mutually independent hamiltonicity of graph G, IHC(G), is the maximum integer k such that for any vertex u of G there exist k-mutually independent hamiltonian cycles ofG starting at u. Inthispaper, we are going to study IHC(G) for the n-dimensional pancake graph P n and the n-dimensional star graph S n . We prove that IHC(P n ) = n - 1 if n ges 4 and IHC(S n ) = n-1 if nges5.