MUTUALLY INDEPENDENT HAMILTONIAN PATHS AND CYCLES IN HYPERCUBES

Abstract

Two hamiltonian paths P1 = 〈v1, v2, …, vn(G) 〉 and P2 = 〈 u1, u2, …, un(G) 〉 of G are independent if v1 = u1, vn(G) = un(G), and vi ≠ ui for 1 < i < n(G). A set of hamiltonian paths {P1, P2, …, Pk} of G are mutually independent if any two different hamiltonian paths in the set are independent. A bipartite graph G is hamiltonian laceable if there exists a hamiltonian path joining any two nodes from different partite sets. A bipartite graph is k-mutually independent hamiltonian laceable if there exist k-mutually independent hamiltonian paths between any two nodes from different partite sets. The mutually independent hamiltonian laceability of bipartite graph G, IHPL(G), is the maximum integer k such that G is k-mutually independent hamiltonian laceable. Let Qn be the n-dimensional hypercube. We prove that IHPL(Qn) = 1 if n ∈ {1,2,3}, and IHPL(Qn) = n - 1 if n ≥ 4. A hamiltonian cycle C of G is described as 〈 u1, u2, …, un(G), u1 〉 to emphasize the order of nodes in C. Thus, u1 is the beginning node and ui is the i-th node in C. Two hamiltonian cycles of G beginning at u, C1 = 〈 v1, v2, …, vn(G), v1 〉 and C2 = 〈 u1, u2, …, un(G), u1 〉, are independent if u = v1 = u1, and vi ≠ ui for 1 < i ≤ n(G). A set of hamiltonian cycles {C1, C2, …, Ck} of G are mutually independent if any two different hamiltonian cycles are independent. The mutually independent hamiltonianicity of graph G, IHC(G), is the maximum integer k such that for any node u of G there exist k-mutually independent hamiltonian cycles of G starting at u. We prove that IHC(Qn) = n - 1 if n ∈ {1,2,3} and IHC(Qn) = n if n ≥ 4.