Mutually independent hamiltonian cycles for the pancake graphs and the star graphs

A hamiltonian cycle C of a graph G is an ordered set 〈u1, u2, . . . , un(G), u1〉 of vertices such that ui 6= uj for i 6= j and ui is adjacent to ui+1 for every i ∈ {1, 2, . . . , n(G) − 1} and un(G) is adjacent to u1, where n(G) is the order of G. The vertex u1 is the starting vertex and ui is the ith vertex of C . Two hamiltonian cycles C1 = 〈u1, u2, . . . , un(G), u1〉 and C2 = 〈v1, v2, . . . , vn(G), v1〉 of G are independent if u1 = v1 and ui 6= vi for every i ∈ {2, 3, . . . , n(G)}. A set of hamiltonian cycles {C1, C2, . . . , Ck} of G is mutually independent if its elements are pairwise independent. Themutually independent hamiltonicity IHC(G) of a graph G is the maximum integer k such that for any vertex u of G there exist kmutually independent hamiltonian cycles of G starting at u. In this paper, the mutually independent hamiltonicity is considered for two families of Cayley graphs, the n-dimensional pancake graphs Pn and the n-dimensional star graphs Sn. It is proven that IHC(P3) = 1, IHC(Pn) = n− 1 if n ≥ 4, IHC(Sn) = n− 2 if n ∈ {3, 4} and IHC(Sn) = n− 1 if n ≥ 5. © 2009 Elsevier B.V. All rights reserved.