On mutually independent hamiltonian paths

Abstract Let P 1 = 〈 v 1 , v 2 , v 3 , … , v n 〉 and P 2 = 〈 u 1 , u 2 , u 3 , … , u n 〉 be two hamiltonian paths of G . We say that P 1 and P 2 are independent if u 1 = v 1 , u n = v n , and u i ≠ v i for 1 i n . We say a set of hamiltonian paths P 1 , P 2 , … , P s 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 e to denote the number of edges in the complement of G . Suppose that G is a graph with e ≤ n − 4 and n ≥ 4 . We prove that there are at least n − 2 − e mutually independent hamiltonian paths between any pair of distinct vertices of G except n = 5 and e = 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 deg G ( u ) + deg G ( v ) − n mutually independent hamiltonian paths between u and v if ( u , v ) ∈ E ( G ) and there are deg G ( u ) + deg G ( v ) − n + 2 mutually independent hamiltonian paths between u and v if otherwise.