Given an undirected graph $G = (V,E)$, a path cover is a set of disjoint paths such that every vertex $v\in V$ belongs to exactly one path. The minimum path cover problem consists of finding a path cover of $G$ having the least number of paths.
One can show the minimum path cover problem is NP-hard by a reduction from Hamiltonian path: a minimum path cover consists of one path if and only if there is a Hamiltonian path in $G$. This reduction is shown in wikipedia.
But is there any result showing the hardness of approximation for minimum path cover in an undirected graph? e.g., what is the best approximation ratio a polynomial-time algorithm can achieve?