On mutually independent hamiltonian paths

Let P1 = 〈v1, v2, v3, . . . , vn〉 and P2 = 〈u1, u2, u3, . . . , un〉 be two hamiltonian paths of G. We say that P1 and P2 are independent if u1 = v1, un = vn , and ui = vi for 1 < i < n. We say a set of hamiltonian paths P1, P2, . . . , Ps of G between two distinct vertices are mutually independent if any two distinct paths in the set are independent. We use n to denote the number of vertices and use e to denote the number of edges in graph G. Moreover, we use ē to denote the number of edges in the complement of G. Suppose that G is a graph with ē ≤ n − 4 and n ≥ 4. We prove that there are at least n − 2 − ē mutually independent hamiltonian paths between any pair of distinct vertices of G except n = 5 and ē = 1. Assume that G is a graph with the degree sum of any two non-adjacent vertices being at least n + 2. Let u and v be any two distinct vertices of G. We prove that there are degG(u) + degG(v) − n mutually independent hamiltonian paths between u and v if (u, v) ∈ E(G) and there are degG(u) + degG(v) − n + 2 mutually independent hamiltonian paths between u and v if otherwise. © 2005 Elsevier Ltd. All rights reserved.