Strongly even-cycle decomposable graphs
  A graph is strongly even-cycle decomposable if the edge set of every subdivision with an even number of edges can be partitioned into cycles of even length. We prove that several fundamental composition operations that preserve the property of being Eulerian also yield strongly even-cycle decomposable graphs. As an easy application of our theorems, we give an exact characterization of the set of strongly even-cycle decomposable cographs. 
Tony Huynh, Andrew D. King, Sang-il Oum, Maryam Verdian-Rizi
math.CO cs.DM