On decomposing even regular multigraphs into small isomorphic trees

It is known that P 4 , the path with 3 edges, decomposes every 6-regular simple graph. It is also known that P 4 decomposes the multigraph obtained by doubling each edge of a cubic graph. We show that P 4 decomposes every 6-regular multigraph with edge multiplicity at most 2. This in turn implies that P 4 decomposes every 6 k -regular multigraph with edge multiplicity at most 2 k . We also investigate decompositions of certain 2 n -regular multigraphs with edge multiplicity at most 2 into double-stars with n edges.