Decomposition of complete bipartite graphs into open trails

It has been showed in [4] that any bipartite graph Ka,b, where a, b are even is decomposable into closed trails of prescribed even lengths. In this article we consider the corresponding question for open trails. We prove a necessary and sufficient condition for graphs Ka,b to be decomposable into edge-disjoint open trails of positive lengths (less than ab) whenever these lengths sum up to the size of the graph Ka,b. Let K ′ a,a := Ka,a− Ia for any 1-factor Ia. We also prove that K ′ a,a for odd a can be decomposed in a similar manner.