What do we know about covering the edges of a graph by disjoint paths? 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?)
You can cover the edges of a graph by cycles iff the graph is Eulerian (or has multiple Eulerian components). Notice that two cycles that share a vertex can always be combined into one cycle.