Why would current flow through at all?
It sounds like the solution you're looking at is wrong, because your point is correct. There is no current through AB and DC. As far as the solution you gave in the comments:


  3A through AB, 4A through DC, 9A through EF and 8A though HG. Batteries w/20V and 15V(in EF) are being charged


Either the solution is wrong or your schematic is wrong. The quickest way to check that is to realize that EF and HG represent essentially identical paths, but your solution has them with different currents.