Note on Integer 4-flows in Graphs

Let G be a bridgeless graph and C be a circuit in G. To find a shorter circuit cover of G, Fan proposed a conjecture that if G/C admits a nowhere-zero 4-flow, then G admits a 4-flow (D, f) such that E(G) E(C) ⊆ supp(f) and |supp(f)∩E(C)|>34|E(C)|, and showed that the conjecture holds if ∣E(C)∣≤ 19 [Combinatorica, 37, 1097–1112 (2017)]. In this paper, we prove that the conjecture holds if ∣E(C)∣≤ 27.