  The paper characterizes $(2t+1)$-regular graphs with circular flow number $2
+ \frac{2}{2t-1}$. For $t=1$ this is Tutte's characterization of cubic graphs
with flow number 4. The class of cubic graphs is the only class of odd regular
graphs where a flow number separates the class 1 graphs from the class 2
graphs. We finally state some conjectures and relate them to existing
flow-conjectures.
