Compatible circuit decompositions of 4-regular graphs

A transition system T of an Eulerian graph G is a family of partitions of the edges incident to each vertex of G into transitions i.e. subsets of size two. A circuit decomposition C of G is compatible with T if no pair of adjacent edges of G is both a transition of T and consecutive in a circuit of C. We give a conjectured characterization of when a 4-regular graph has a transition system which admits no compatible circuit decomposition. We show that our conjecture is equivalent to the statement that the complete graph on five vertices and the graph with one vertex and two loops are the only essentially 6-edge-connected 4-regular graphs which have a transition system which admits no compatible circuit decomposition. In addition, we show that our conjecture would imply the Circuit Double Cover Conjecture.