Compatible Cycle Decomposition of bad K5-minor-free graphs

Let G be an eulerian graph. For each vertex v∈V(G), let T(v) be a non-empty subset of a partition of the edges incident with v into 2-subsets and set T=∪v∈V(G)T(v), called a transition system of G. A transition system T of G is admissible if T∩F|≤12|F| for every T∈T and every edge cut F of G. A cycle decomposition C of G is called a compatible cycle decomposition (CCD for short) of (G, T) if |E(C)∩T|≤1 for every cycle C∈C and every T∈T. H. Fleischner proved that if G is planar, then for every admissible transition system T of G, (G, T) has a CCD. G. Fan and C.-Q. Zhang (2000, J. Combin. Theory Ser. B 78, 1-23) showed that this result is also true for K5-minor-free graphs. We generalize this result to all eulerian graphs that do not contain a special type of K5-minor which is called a bad K5-minor. To this purpose, we characterise the 4−regular “hereditary” bad K5-minor-free graphs (G, T) in which every CCD of (G, T) is a pair of hamiltonian cycles.