The Circular Flow Number of a 6-Edge Connected Graph is Less Than Four

We show that every 6-edge connected graph admits a circulation whose range lies in the interval 1; 3). The circular ow number c (G) of a ((nite) graph G is deened by c (G) = inffr 2 R: some orientation ~ G admits a circulation f : E(~ G) ! 1; r ? 1]g: This parameter is a reenement of the well studied ow number (G) := d c (G)e, which was introduced by Tutte as a dual to the chromatic number. Since c (G) 2 with equality if and only if G is eulerian, the circular ow number may be regarded as a measure of how close a graph is to being eulerian. The following results and conjectures can be found in 4].