Two related things I have heard/know of are,
(1) That there exists a polynomial algorithm to find a cover of the vertices by $k$ vertex disjoint cycles. (Can someone give a reference for this?)
(2) That its NP-complete to decide if $k$ vertex disjoint paths can cover all the vertices of a graph.
Are there edge analogues of these results?
Does using the dual graph translate these results into their edge versions?
If the line graph is a edge-to-vertex-dual then is there a vertex-to-edge-dual? (which will may be convert the question of finding a vertex cover by k vertex disjoint paths into a question of finding an edge cover by $k$ edge disjoint paths?)