The difference between the circular and the integer flow number of bidirected graphs

Let F c ( G ) and F ( G ) be the circular and the integer flow number of a flow-admissible bidirected graph G , respectively. Raspaud and Zhu proved that F ( G ) ? 2 ? F c ( G ) ? - 1 . This note shows that this result cannot be improved. Moreover, in the same paper, Raspaud and Zhu conjectured that F ( G ) - F c ( G ) < 1 for every flow-admissible bidirected graph G . This conjecture was disproved by Schubert and Steffen, who showed that ? F ? 2 , where ? F = sup { F ( G ) - F c ( G ) : G ?is?a?flow-admissible?bidirected?graph } . Our result implies that ? F ? 3 . Furthermore, if Bouchet's 6-flow conjecture is true, then ? F = 3 .