A note on range-restricted circuit covers

Let Cone(G), Int.Cone(G) and Lat(G) be the cone, the integer cone and the lattice of the incidence vectors of the circuits of graph G. A good range is a set K of natural numbers such that the intersection of Cone(G), Lat(G) and K^E is contained in Int.Cone(G) for every graph G=(V,E). We give a counterexample to a conjecture of Goddyn [ Goddyn, L.A.: Cones, Lattices and Hilbert Bases of Circuits and Perfect Matchings. Contemporary Mathematics 147, 419--439 (1993)] stating that N\{1} is a good range.