Arbitrary decompositions into open and closed trails

The problem of arbitrary decomposition of a graph G into closed trails i.e. a decomposition into closed trails of prescribed lengths summing up to the size of the graph G was first considered in the case of the complete graph G = Kn (for odd n) in connection with vertexdistinguishing coloring of the union of cycles. Next, the same problem was investigated for other families of graphs. In this paper we consider a more general problem: arbitrary decomposition of a graph into open and closed trails. Our results generalize all known results on decomposition of a graph into closed trails as well as some results concerning decomposition of a graph into open trails.