Nearly nowhere-zero -flow graphs

Let F c ( G ) denote the circular flow number of a graph G , and r be a rational number. This note proves the following statement: Let G be a k -edge connected graph with k ≥ 2 . (1) If G has an edge e such that F c ( G − e ) ≤ ( 1 − 1 / k ) r , then F c ( G ) ≤ r . (2) If r ≥ 3 is an integer and G has an edge e such that F c ( G − e ) < r , then F c ( G ) ≤ r .