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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 useē 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 −ē mutually independent hamiltonian paths between any pair of distinct vertices of G except n = 5 andē = 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. Definitions and notationFor the graph definition and notation we follow u, v) is an unordered pair of V }. We say that V is the vertex set and E is the * Corresponding author.