Even cycle decompositions of 4-regular graphs and line graphs

An even cycle decomposition of a graph is a partition of its edge into even cycles. We first give some results on the existence of even cycle decomposition in general 4-regular graphs, showing that K5K5 is not the only graph in this class without such a decomposition.Motivated by connections to the cycle double cover conjecture we go on to consider even cycle decompositions of line graphs of 2-connected cubic graphs. We conjecture that in this class even cycle decompositions always exists and prove the conjecture for cubic graphs with oddness at most 2. We also discuss even cycle double covers of cubic graphs.