Decomposition of complete bipartite graphs into open trails

It has been showed in [Horňák, M., and M. Woźniak, Decomposition of complete bipartite even graphs into closed trails, Czechoslovak Mathematical Journal 128 (2003) 127-134] that any bipartite graph K a , 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 show that for even a and b any complete bipartite graph K a , b is decomposable into open trails of arbitrarily lengths (less than ab) whenever these lengths sum up to the size of the graph K a , b . Let K a , a ′ : = K a , a − I a for any 1-factor I a . We also prove that for odd a K a , a ′ can be decomposed in a similar manner.