On mutually independent hamiltonian paths

Let $P_1=〈v_1,v_2,v_3,\dots ,v_n〉$ and $P_2=〈u_1,u_2,u_3,\dots ,u_n〉$ 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 $u_i\ne v_i$ for $1<i<n$. We say a set of hamiltonian paths $P_1,P_2,\dots ,P_s$ 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 $\stackrel{?}{e}$ to denote the number of edges in the complement of $G$. Suppose that $G$ is a graph with $\stackrel{?}{e}\le n?4$ and $n\ge 4$. We prove that there are at least $n?2?\stackrel{?}{e}$ mutually independent hamiltonian paths between any pair of distinct vertices of $G$ except $n=5$ and $\stackrel{?}{e}=1$. 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 ${deg}_G\left(u\right)+{deg}_G\left(v\right)?n$ mutually independent hamiltonian paths between $u$ and $v$ if $(u,v)\in E\left(G\right)$ and there are ${deg}_G\left(u\right)+{deg}_G\left(v\right)?n+2$ mutually independent hamiltonian paths between $u$ and $v$ if otherwise.