Oriented Circuit Double Cover and Circular Flow and Colouring

For a set C of directed circuits of a graph G that form an oriented circuit double cover, we denote by IC the graph with vertex set C, in which two circuits C and C are connected by k edges if |C ∩ C| = k. Let Φ∗c (G) = minχc(IC), where the minimum is taken over all the oriented circuit double covers of G. It is easy to show that for any graph G, Φc(G) ≤ Φ ∗ c (G). On the other hand, it follows from well-known results that for any integer 2 ≤ k ≤ 4, Φ∗c(G) ≤ k if and only if Φc(G) ≤ k; for any integer k ≥ 1, Φ∗c (G) ≤ 2 + 1 k if and only if Φc(G) ≤ 2 + 1 k . This papers proves that for any rational number 2 ≤ r ≤ 5 there exists a graph G for which Φ∗c(G) = Φc(G) = r. We also show that there are graphs G for which Φc(G) < Φ ∗ c(G).